I'd like to put a small seat somewhere in our shower. Is this
something I might be able to do myself or should I hire a contractor?

I'm thinking of something that I can just screw or glue onto the
existing shower. I don't want to remove any tiles and build it in. Is
that possible or reasonable?

The shower has 4x4 tiles over sheetrock.

Here are a couple of examples I found.

http://www.johnbridge.com/how-to/shower-seat/

http://tinyurl.com/5vbg3rk

It doesn't need to be as fancy as either of those.