DISCLAIMER: My sources are not all scholarly, but I'm assuming the information provided is accurate for the sake of my answer. I'm also not a physicist. Take these answers with a rather large grain of salt.
With that out of the way, let me try my best to answer this...
First, let's take a look at interstellar medium. If there's (MAX) .1 particles per cubic centimeter, and 75% of the medium is hydrogen, that gives us .075 particles per cubic centimeter. And that's probably a a high estimate. So you then convert 1% to antimatter. That gives us... a whopping .00075 particles per cubic centimeter. That doesn't sound like much, but it adds up fast. That's 750 useful particles per cubic meter, which is quite a lot! I'm totally ignoring all the problems with just holding on to antimatter for long enough to use it (check out the Atomic Rockets page linked below if you're curious)! Also, I don't really know what to do with this information, but it may be useful if combined with more technical information about your rocket engine.
OK so there's a lot of different ways to harness this antimatter, but none of them are a ramjet, apparently because "antimatter ramjet" is not something that really exists, since ramjets and antimatter don't mix well. Oh well. Apparently Michio Kaku covers this in his book Physics of the Future (found here on amazon) if you're curious and really want to dive deep into this. That's kind of the end of my formal answer to your question, but read on if you want some theorycrafting and more guesswork.
Relatedly, you're going to need some pretty darn thick shielding, since hydrogen's antimatter pions are remarkably annoying as far as trying to murder you goes. Additionally, you might be better off trying some other material, since Nitrogen, for example, has a charged pion to propellant efficiency of 95%
The Atomic Rockets page has a "Antimatter Rocket Equation", and so I'll just use that for fun to see if you really could accelerate at 1 g. Well, here's the thing- the equation doesn't use the antimatter as reaction mass. Oh well. Back to the internet drawing board. (There is a group of wonderful charts for antimatter reactant for different propulsion speeds but quite honestly I'm not sure that helps you, since you're looking for acceleration, not delta-v. Onwards and upwards...)
My quest ends here. I'd highly recommend you check out Atomic Rockets, it's a wonderful source for all things sci-fi. I was unfortunately unable to find anything calculated for antimatter ramjets, so I'd definitely look more into that. I did, however, find an article claiming 1 g of acceleration using RAIR (that's the "nextbigfuture" link if you're curious), and the paper cited in the article may provide some answers. I didn't read the whole thing, but from my skim I didn't notice any glaringly useful information. It also details a sort of mixing antimatter in with other reactant, so that's a little bit different from your engine.
This turned into an absolute mess of an answer, but I hope that provides at least a little bit more information for you, and I wish you the best of luck in your spaceship design process!