My first thought is the V-22 Osprey. Being a real aircraft, you can find specs on Wikipedia and use that as a base for your design. It's VTOL, it's fairly fast (much faster than an airship), it's designed to cargo stuff around, and you can add weapons to it, so that's a good candidate for a cool, real-ish, flying base for a squad of image-conscious heroes.
You shouldn't expect a palace though. It's conceivable that you can build a flying tour bus for a handful of people, but it'll be a tight fit and you won't have much room for cargo inside (although you can still tow cargo outside). That said, big roomy coolcraft doesn't sound very cyberpunk to me, so maybe that's a bonus. And on the flip side, you can land it pretty much everywhere and camp in the great outdoors, and that'll be more room than any sci-fi spaceship can give you.
The question is tagged "engineering" but...
The problem isn't engineering, it's logistics
And even then,
The real problem is fuel
Hands down, how you're refueling your plane will be the single biggest design decision to this question. I say that because it doesn't just shape the plane, it shapes the world and the plot.
Refueling on land
The basic option is to land and go to the gas station to get your kerosene. The first limitation here is landing somewhere that has the infrastructure and that welcomes you.
Fiction gives you the opportunity to place private and/or pirate airfields all around the world. Private airfields means you'll have to ask permission, book a slot and pay for it. Pirate airfields might not bother with such formalities, which is also a great way to induce shenanigans, like that time you landed in Ulaanbaatar and pirates kidnapped your pilot in broad daylight.
You can also land anywhere in the plains of Africa or the steppes of Asia without being bothered by the neighbours and set up shop in the wild. That's good to do some basic maintenance, and if you can source some fuel from there it's a valid option, though you obviously lose the benefits of an established infrastructure.
The other major limitation here is that when doing a Reykjavik-Jakarta to get to your next mission you'll need to land once, twice, or maybe more. That means the problems above become recurring, and ultimately it's going to eat some precious time you might not be able to afford, especially if aforementionned shenanigans occur. Whether that's a pro or a con for your story is up to you.
At any rate, if you go for that option, a straight-up Osprey-type aircraft will do the job just fine.
Refueling in the air
We do have the technology to bring the gas station directly to you in-flight. The problem is you know that's stupid expensive because no commercial airlines do it, only the military can justify that spending somehow. A cyberpunk setting might allow you to introduce a commercial service of tankers to refuel private citizens such as your heroes. But that means flying an air-bus around for 'round-the-world trips is a common enough occurence that there's a market for it.
That's a pretty big condition, and one that will potentially reshape your world. If refueling in-flight is widely available, your heroes are more likely to get contested by air-pirates, corporations, and private militaries having their own air-bases. If everybody has a cool air-base, your own cool air-base may lose its romanticism.
Maybe the biggest limitation here is the stupid expensive part. Unless your team is swimming in cyberdollars, such a refuel might be more than they can afford. And that means they'll have to hijack a tanker or two. While it is a pretty cool image, it brings a lot of complications to your travels. Your plane will need to be designed for combat, and obviously that means you should expect to spend a few chapters fighting for fuel at 30000 feet.
If you aren't interested in a world that's highly militarised, with weaponising your plane, or if risking your life for every refuel sounds too big of a hassle, you might consider this a bad option.
Not refueling, like, at all
The most realistic alternative would be solar power. Solar power is nice in that you might be able to fly indefinitely for cheap. You do have to account for daylight and the size of your batteries when planning your route. In some cases, that might be very inconvenient, but still, solar energy is cheap and readily available most of the time. It gives you the opportunity to stay in the air much longer without needing a refuel.
Of course, your setting might have a better alternative. Whatever your energy source ends up being, it needs to be:
- light,
- compact,
- powerful,
- have enough capacity to keep you in the air for a convenient amount of time.
Ideas may include a pocket nuclear reactor. Nuclear energy offers a lot of opportunities for shenanigans.
You might also think of that refuels by sucking gases the atmosphere. For an added bonus, you can make it something found at a specific altitude. Imagine your aircraft needs a specific mix of hydrogen-2 and helium-7 that's only found just at the operational ceiling. And this is just an example with zero scientific backing, but if you establish it with enough confidence, it'll play.
Additional consideration
An aircraft needs space to fly. You're thinking "not a problem, the sky is full of that", but the sky also belongs to somebody. It's one thing to fly a drone or a Cessna without authorisation, but governments/corporations will notice a big military plane flying around the world and might take offence if it goes through their sovereign airspace.
There is plenty of airpsace where nobody will come bother you. The middle of the ocean for instance, and the great deserts of the world. For everywhere else, you can embrace piracy with an IDGAF attitude (which is fairly cyberpunk), and hope you can dodge the jet fighters indefinitely. You can also add everybody's favourite part of a story: bureaucracy and red tape. Or you can do both: violating airspace of people with poor radar coverage or weak air forces, and file the paperwork where you are likely to get shot down on sight.
That gives you the opportunity to hack the mainframe to give your aircraft the right credentials. That also gives you the opportunity to get caught with fake credentials and having to hightail it out of missile range.
Which is to say your aircraft will be designed for speed. You might want to replace propellers with jets for more speed, and also for more coolness. That should give you a better chance when you need to make a quick exit.
I wouldn't bother with stealth. It only helps to the point where nobody looks up and sees your big airplane hovering above the headquarters of Megacorp, or if nobody can hear the roar of the engines from over the horizon.
In conclusion
The only question you need to ask yourself is what fuel you put in the tank, and how you get that fuel.
The rest has largely been solved by real engineers in the real world, which is to say your cool aircraft looks like an Osprey. It can be bigger, it can be haphazardly converted to solar power, and maybe you have tiltjets rather than tiltrotors, but a big VTOL is just about the gist of it.
