Forget about the air compressor for a moment. You should really reconsider getting the nailgun at all.
First off, in your specific project, that nail gun is not going to do the job. The one linked and those like it are for putting nails into wood, they won't put nails into concrete. You try and shoot a nail into concrete or stone with one of those, you are going to be in for a very unpleasant and dangerous surprise.
Even ignoring that, framing nailers are not a very good use of money for DIY usage. I have one, and it very rarely ever gets any use. Even when I do use it, I use it more to rationalize hanging onto it, every time I use it I could also use screws just fine. Especially considering the fact that you are considering one for the wrong task, it is especially clear you don't need it. The real advantage of a nailgun is that nails are cheap and the gun is fast, but for DIY, neither of those criteria are important. Instead, just use construction screws with a regular drill. Buy a few boxes of the DeckMate self-tapping screws and you're good to go, those things are awesome.
Sure, nailguns are big and loud and fun to machine gun nails into things, so if you really want to use one because of that, rent one first. After you've seen how (in)frequently you need one, then go back to the decision of whether to buy one.
For attaching your framing to the concrete, use a hammer drill and appropriate concrete screws. Buy or rent one as appropriate. Hammer drills suitable for drilling a handful of screws for your project have a great deal of additional utility vs a framing nailer which is really only good for rough carpentry.
Lastly, a compressor like that is fine for powering a nailgun like you linked for DIY purposes. The specs call for a bigger compressor, but that's for construction crews that are shooting thousands of nails a week and are getting paid by the hour. A small compressor you may have to wait occasionally for the compressor to recharge, but for DIY that's no big deal. Just don't try to shoot nails into concrete!