For the means you have listed you could just get a 16gb iPod touch 4 for $200 and save $100
-Music and podcasts will be supported as long as iTunes sells music.
-Web browsing is pretty fast on the iPod touch 4, so is email. I have no problem with mine.
-Movies, with the 3.5 inch retina display its fine to watch movies on it. Only get the 5th gen if you want to watch it in a bigger screen.
-Syncing Documents will easily be done with the iPod touch 4.
Really you shouldn't spend $300 on the new iPod touch when for what your doing the iPod touch 4 will do fine. And you can get some nice accessories from that $100 saved.
Also side note, games work fine on my iPod. Even Real Racing 2 works great on it. No crashes from my device. You didn't list camera so I assume you won't mind the iPod touch 4's horrible camera.
Only reason to get 5th gen is... Faster Processor (A4 still does fine to), Siri (Not needed for your needs), more iOS 6 features (Like 3D maps, again not needed from you), a new design, 5MP camera (You don't say anything about taking pictures so its not needed), and a silly loop thing

.
Overall I recommend the iPod touch 4 for your needs. Save that $100!!
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Yes, spend the extra $50 and get the new 5th Gen iPod Touch.
Think about it like this, in a year, iOS7 will release and Apple will most likely cut off support for the 4th Gen Touch. The A4 chip in the 4th gen is really outdated now, and the lack of RAM means it'll run slower with lag.
For an extra $50, it's worth it for the 4" screen, A5 chip, and an increase in RAM. Everything will run smoother and you know you're guaranteed to get updates for the next 2 years at least.
I don't think the OP really needs iOS 7 for his needs. I also don't think that for what he is doing the A5 chip and more RAM will make much of a difference.
AND the iPod touch 4 16gb is only $200.
EDIT: Disregard this if you are getting the 32gig model.