Hello, I am planning to get a Intel i5 3350P for my budget gaming build, and I am wondering if it will do its job well for games. Also, I'm pairing it up with a 7870, if I wanted to upgrade later, would it bottleneck anything?
The I5 3550P will be absolutely fine. It's one of the best gaming CPUs and there's no way it could possibly bottleneck that graphics card. Even if you ran 2 7870's in crossfire, they would be the bottleneck, not the I5.
Personally, I'd go for the 3470/3450 over the 3350P as the entry level i5. It's got the same clockspeed as the 3450 but a worse turbo mode. Depending on where you live, they are usually priced similarly.
No modern i5 is likely to cause any kind of bottleneck anyway.