This just happened to me. Mines a year and a half old as well and 4th generation. I was looking on Google and that didn't really come up with a solution other than its been happening to a lot of 4th generation iPods. I tried using some cleaner to help remove the dust/dirt but it only made things worse. Better for about 4 clicks then back to not working. If I find a solution other than taking it to Apple I'll be sure to tell you.
My brother screwed up my iPod by getting it wet on a sink like an idiot, THEN I put it in rice to dry it out, he had the chutzpah to take it out before the required time. It worked, but the backlight was broken, also the home button didn't work, and in March, it would stop syncing music and the WiFi didn't work. What I had to do was only go to music, as that's ALL I used it for. Does the reset button work? If it does, use it whenever ya wanna return home, atleast until your button is fixed.
I've heard that a single drop of rubbing alcohol and repeatedly pressing the button will help to remove anything stuck under there. That said, that will void your warranty. That's probably your best bet without spending any money at all.
Unless you have AppleCare...in which case go to an Apple store. Actually, go to an Apple store either way and ask. Worst that happens is that they say no sometimes they're strangely generous with fixing/replacing stuff that's out of warranty.
I've tried the alcohol a few times, but it doesn't help. Should I keep trying or just wait until I can bring it to an Apple Store? I have another iPod here (Totally busted 2nd or 3rd gen with severe water damage, was a friend's) with the part I need still intact... Do you think they would put that part into my 4th generation?
No, they won't do the repair for you. Also, the parts from the 2nd and 3rd gen won't fit in the 4th gen.
That said, the new buttons are really cheap. It might not cost much for you to get the ipod fixed. I know you didn't want to spend money but you're going to have to if you want to get it fixed. The alcohol thing was really your best and only bet on that front. I'm sorry I wouldn't keep trying it, it's probably not going to help