I have to agree with JustinKlemgold.
By all means you can turbo an automatic. You wouldn't be the first or last.
But, the moment you start running that power through the auto tranny is the moment you start wearing it out. Sure you can beef up the transmission, but that will cost as much if not more than a turbo setup.
If you want cheap and easy power upgrades, then do intake, headers, and exhaust. If you want a car that's even more fun to drive, either swap in a manual transmission, or buy an integra that was already a manual. If you don't have the tools or space to do the work, buying one with a manual will be your best bet unfortunately.
After that, you can do intake, headers, and exhaust (or maybe a manual integra on the market in your area already has these upgrades).
If you want some more power upgrades, you will have to look into internals, and/or go for a nice turbo setup.
Personally I would love to try an auto turbo integra, that sounds awesome