Living in a small apartment - strike one
A Behringer mixer and an SM58 - strike two
Waiting on a condenser - strike three
Wrong forum for these types of questions - strike four (I'll take this one back since Cake moved the thread)
Carpet on the outside of the booth - strike five
I have a suggestion. And this is based from years of experience and having been there and done that. Forget building a vocal booth. Save your money for that condenser mic and maybe even a better mixer or vocal channel. Instead, go upstairs and knock on your neighbors door and tell them that occasionally you will be recording vocals in your apartment and that they should not call the cops on you (which immediately puts you at risk in the event someone should break into your apartment and try to rob you). Then get yourself some heavy duty moving blankets, pillows, sleeping bags, whatever and strategically postion them in your bedroom or wherever you are recording.
As Mike already pointed out, what you are sort of alluding to is an airtight chamber. You can actually buy prefabed vocal booths but they are expensive. I suspect, even if the thought crossed your mind that you would somehow get the lumber and attempt to build one in your small apartment, the best you could hope for is that it would cut down some of the noise but not all of it. You will find it cramped and uncomfortable. Without some sort of airflow/air conditioning, you won't be able to spend more than a few minutes at a time in there.
Then, to add insult to injury, you will have to dismantle this thing when it comes time to move. Do yourself a favor and make the best of it with hanging blankets and such that can be easily put up and taken down. Talk to the neighbors. If none of that works, move.
Best of luck - let us know how it turns out!