My vote is for maybe realistic, with some tweaking (either of abilities or expectations).
Physically, his abilities would have to be top notch, equivalent to Olympic-level athletes - although not actually an Olympic athlete, since that requires a lot more specialization to the point it might be counterproductive, but the raw abilities of one. If we want to be specific, perhaps the abilities of an Olympic-level acrobat or gymnast, since the flexibility, control, and body-awareness will be very useful (physically specializing towards, say, strength or sprinting-speed might not be versatile enough). There will be people faster, and stronger, and people with more specialized training in a single area, but if he has a decent level of native skill and takes his training seriously, he can maneuver better, fight where he can and out-think, plan traps, or manipulate the situation where he can't.
As for age, physical fitness tends to peak in from the 20s to early 30s. If he starts his 'career' at 20, or maybe even a little younger, say 18, he has at least a decade and a bit with which to pick up the experience and skill he will need to be able to compensate for the effects of aging and also the accumulating injuries form fighting, and keep his career going a little longer. Even so, his age will be a limiting factor - there will be no 60-years in this career no matter what the comics say, 3-5 years is reasonable career length for sports (given chance of injury) and most athletic careers are over by age 33. Even if he can use experience and skill to compensate - by stacking the deck, choosing fights wisely, being sneaky and underhanded, and not directly competing where he doesn't have an advantage, at some point he will no longer be effective in direct fights against younger, stronger foes even if they are less skilled. At that point, he might be able to train a successor and/or serve in a support capacity if he would like the name to continue longer, but his career is all but over.
He probably won't be a "master" of all martial arts, that seems a little too much - especially since they will be specialized and mastery of each will require different strengths and weaknesses. But basically familiar with many types, sure - and a master (or nearly so) of two or perhaps three different styles might pull of the fighting quite well. The muscular control and body-awareness that comes with mastering a style or two might let him fight quite effectively even if not in his chosen style, if a specific style (that he still has to be familiar with) happens to be the best match for an opponent's style. And given the raw, Olympic level abilities, and starting training young and with a lot of motivation, being somewhere from fairly good to master-level in perhaps two martial arts might be reasonable when just starting out. Picking up the basics of other styles (and maybe mastering a third, depending) would be less critical, and could happen at any point along the way. That is, they would help with versatility and flexibility, but they would also take extra time and effort.
As for wounds and injuries, this will need some lowered expectations. He can be skilled, and careful, and have the best in body armor and gear - but injuries will still happen, and will not be magically healed. Stories are usually pretty bad at this, since they want shock value without actual consequences. So, every time he gets injured, he will need time off to heal, and he will need time off for bullet wounds, for broken bones, and also for strains and sprains, burns and cuts, and even significant bruising, since he needs to be in fighting condition. He will need periodic time off anyway, recover from strain, stress, lack of sleep, to catch up on all the littler healing (scratches and bruises and and whatnot) since that will interfere with him fighting...and also to basically keep his life balanced and livable. It is possible that this time happens, but "off-screen" in the media version, and he could do non-physical work in the meantime (perhaps also "off-screen" extra time investigating).
But in general, he will probably not be out nightly but maybe have appearances spaced over a month, or even a year. If his costumed appearances are tightly correlated to injuries, it will also make it much easier to track his identity - anyone injured during relevant time periods, over time, will probably give a fairly short list, with the added bonus of going out while injured/recovering might also let people figure out and track injuries quite specifically (still favoring that knee, something with the shoulder, etc). On the other hand, if his appearances are irregular to begin with, that might mask healing time between regular investigation-time and non-costumed activities.
Mentally, he might not need quite so impressive abilities to be quite effective. He will need to be highly intelligent, yes, and have a good memory, a decent problem-solving ability, deductive reasoning, and the ability to think unconventionally and look for out-of-the-box solutions. He will need a decent level of ability to figure out things on the go, while they're happening, and to change his plans to changing circumstances. And obviously the more native abilities he has, the easier all that will be. But really, he doesn't need to have abilities or training out of reach for a very good detective or investigator. And, studying and practicing can improve native abilities quite a bit - not to mention, he can lean on artificial help, in terms of computers, gadgets, and programs, to achieve quite a bit more.
So, for the rest - he doesn't need to memorize tons of details with access to a computer with which to look them up, maybe just things he expects to need on a day-to-day basis, or what he needs for a single outing. If he's got a decent connection to that computer on the go (I think I recall a glove-computer or the like), he can look things up if he has even a few minutes to stop and check. Or a (good) voice system so he can ask the computer to look stuff up for him if he hasn't got his hands free. Or a single accomplice, who can just access the computer and do a search if it turns out he needs more information between going out and coming back in. If he has a little camera on his suit, or audio recording, he only need a decent memory, not a perfect one - he can look over recordings after the fact to pick up details he might have missed. With money comes the potential to buy or access copies of the best investigation programs, from pattern analysis to forensics to tracking programs.
But wait, if he has the mental abilities, and resources, of most pretty good detectives or investigators, how does he continuously out-perform, for example, the police? Well, he can pick and choose which cases to investigate, while actual police (and so on) are investigating many cases at the same time, since they are responsible for all of them. If he has too much going on to deal with some crime, he can drop off tips and information to the cops instead of investigating everything himself. Also, between having lots of money and only needing enough resources for himself, he can always have the best equipment and most cutting edge programs, while police and so on will settle for the most cost-effective and still pretty good resources, and will have to share them among many people and cases. So not only can he focus in on just whatever he happens to be working on, he can also spend a long time going back to a case until it's solved to his satisfaction, without the outside pressures (the media, the higher-ups, the politicians) that the police might have to deal with on any given case.
The more realistic version will probably need to spend more time analyzing and researching the crimes, compared to the media version whose investigation time would be truncated or "off-screen", but with a tighter focus and the best of resources, he could still be very impressive. And again, the more time between appearances will let his healing and recovery time drop to a more realistic level.
Hacking and programming, hm. I think he would need to be quite good, to make it work - but again, while the better his skills the easier, he doesn't have to be genius level to make it work. He can augment his actual skills with access to the best programs and tools for the purpose, including those used by police (for breaking into digital systems) or computer repair places (for data recovery and such), but he will want to be fairly good in his own right for situations where he is without the regular tools. It would be easier if his company hired really good computer people and did some work in the relevant areas, had some division for developing such programs for law enforcement, (which will also give him a cover for procuring his own copy of the latest).
Or at the very least his company should be pretty careful about cyber-security and he keeps himself up-to-date on what they were looking out for, borrowing their expertise to figure out exactly what he needed to learn and what he could get around other cyber-security systems. He would need to know enough to take what they were doing and apply it elsewhere, but he could piggyback off of their results instead of having to be a genius about it himself. Having even one trustworthy computer-person in the know would help a lot, but it might be worked around. It also might be helpful, in a different way, if his company was doing work on computers instead, and thus he could (sneakily) have some back door installed (for legitimate, um, data-recovery purposes!) that would let him have a shortcut if whatever computer he needed to break into happened to be of his company's make.
And speaking of which, logistics. Obviously, if he spends company resources on developing the gadgets (and/or producing them) and then doesn't sell them, his company will want to know what the heck is going on with R&D. On the other hand, it might not be a bad idea to produce and sell some of the gadgets for use, the gadgets that are more useful than dangerous, anyway. The gadgets being sold might be larger or less efficient or less tricked out versions anyway (due to pricing or ease of use, or deliberate choice). Perhaps the "best" version would sometimes still be sold for military/police use, as well - it gives a reason for still making them, and if they are available to buy, it is harder to point fingers at who might have them than if they are actually restricted. He should probably "buy" them from the company, though, at least replace enough of the cost from his personal fortune, to not run up red flags about someone skimming profits. For anything he reasonably can, he should be buying the best products available (by proxy, if possible), and maybe modifying them as needed for his specific use... the fewer things he has that aren't widely available, the harder it is to track him down using them.
Other gadgets, ones more specific and less hide-able, he might want to make or acquire equipment for making himself. Most rarely, he might develop or partially develop a gadget through the company, and never put it into production - something which will cause questions if he does it too much, but might work if really needed. For batarangs, or whatever, a machine to produce specific shapes from metal might be a good investment (since he needs a lot of them, and I think they're visually distinct), and also producing solid metal cutouts might not be too complicated of a process. Or at the least, he can acquire such machinery for (or from) his company, if he can just hide the specific mold or equivalent that is needed to actually make them. For other gadgets, he might be able to acquire needed parts from piecemeal from other companies and spend time assembling or modifying them - at the cost of time and for the value of secrecy. If he's careful with the equipment, and doesn't loose it all over the place, he might be able to get away with a few gadgets not market-available without people knowing enough about them to try and track back to the source.
As for economics, a billionaire's budget should cover most of what I've mentioned (programs, really good computer system for one, some gadgets), as long as the billions are his personal fortune, not his company's - although, if he doesn't want to be known as openly funding, well, himself, he might want to combine frugal personal spending habits and a public image of a spendthrift, so people don't wonder where his money is going. If he has more money tied up in his company, it will be trickier - since budgeting, or auditing or internal checks (for tax purposes, for internal budgeting, to prevent corruption) might turn up cash flow discrepancies, that may look like anything from embezzlement to money laundering to financing something underhanded (which I guess this is). If at all possible it will be better to keep the business as clean as possible, paying the company for raw materials or damages and keeping the cash flow clean.
Lastly, social media. This will be trickiest, and the least plausible answer. Social media is everywhere, and between camera-phones and tracking software and people, oh, people everywhere, keeping something like this secret is going to be very difficult. To start with, he will need a body double in the know, or several who may have different levels from being in-the-know to being ignorant. Because people will ask questions if he needs to drop out of sight, and announces (say) a trip, and there's no reports back of sightings in the areas he was "going to". If one person covers too many of these trips, they will likely get suspicious very quickly, so they must be somehow in the know. On the other hand, if there are several, using them at different times, each with some semi-plausible excuse for missing a single event or trip and not wanting to admit it, may cover longer stretches with no single one getting suspicious about the accumulated time - as long as they are somehow prevented from comparing stories.
Beyond that, any visible injuries will be dangerous, since they will likely be recorded (and available for comparison) long after they have healed. They can build up a case over time, especially if compared against injuries known to be suffered while in costume - because once or twice may be coincidence, but over time people will get more and more suspicious. And they may also reveal the body-double scheme if not handled carefully. Also, he will need a lot of really good stories for the injuries and scars, because not only will they accumulate very quickly, and leave distinctive marks (wait, how many bullet wounds?), they are likely to be continually renewed, from very old to barely healed - so can't be attributed to a single, or even a few, larger events instead of continual injuries.
It will be hard to explain why he doesn't stop such activities that dangerous or beyond his skill level even long enough to heal, and there won't be official backup for any such story (police or hospital records, or reports of similar accidents or injuries) when people do get curious, and look for answers. They might assume scandals, they might assume criminal activity (either towards him, like abuse, or from him - which vigilantism kinda is) - but once people start investigating, those discrepancies will stand out.
So, to make it work... his daylight persona also can't be out and about every day, and will need to be able to drop out of sight sometimes with little notice. He might be able to be out more during the day if he goes out at night less (when injured, or when still investigating, or during down-times or such), but since going out when injured is not a great idea unless the injuries can be very well hidden or attributed, less exposure is better. He will need body-doubles, and fair reasons for them to believe that he wants to duck out of whatever and not tell why (which might involve "fake" secrets, but whatever). He will need his doctor to be in on the secret (and also capable from primary care to serious patching-up), and willing to spread misinformation on his behalf when people come questioning.
For his helpers and those in the know, take your pick of who is in the "five or less" the question asks for - at least one general helper for computer access while out, basic patching up, organization, and giving excuses about him; his doctor, for larger injuries and also primary care; some accomplice in his accounting or business who knows what is really going on with the cash flow and why, to misdirect others; helpfully but not required - a really good computer person who can check on the computer work, help with programs and such; helpfully but not required - at least one body double so they aren't asking questions about why he needs to be out of sight so very often; and probably (depending on circumstance), whoever trained him might have some idea of what he was doing with that training even if they aren't "actively" helping, especially investigation training and martial arts - the former has few other uses, especially to be good, and the latter might identify him from familiarity with fighting style and/or seeing a fight in costume.
So, to sum up - the more realistic version will go out much less often, and spend extra time being his cover persona, researching, healing injuries, and acquiring, customizing, upgrading and fixing the gadgets and programs he uses. However, his ability to focus in and spend the time and effort beforehand means he will be quite effective on those occasions where he is out and about, and irregular appearances will let his reputation have time to grow in the telling. He needs to be decently wealthy, and his company should at minimum have a division doing R&D for law enforcement or military tech - including programs, tools, and forensics, so that he can legitimately acquire and develop the things he needs and even profit from it, though much of what he uses (or is seen to use) should be commercially available, so it is harder to track him that way. He needs to openly claim to practice/enjoy something dangerous enough to justify the continual injuries, and also need to be personally unreliable, and have enough reasons to be that way, to justify body-doubles. He has three to five accomplices, and possibly two or three other people who might guess, given the right circumstances (but who also might listen to the reasons why and be silent). He probably should start late teens to early twenties, he can count on his physical prowess for no longer than ten years from that, and he might stretch his career a little further through experience, underhanded tactics and stacking the deck - though perhaps not much further, before he has to find a successor if he wants the name to continue. At which point he can be one of the five support people for the next one. I think that's most of it, really.