That PSU is too small for the HD 7870. It only has a 22A +12V rail. It is only the equivalent of a 300W 'real' PSU. Some manufacturers use "poetic license" to rate their power supplies. You need a 'real' 600W PSU to move up to that card. But your motherboard seems to be a 1155 socket if you now have a Pentium G2020. So you should be able to move to the i5.
But whichever card you end up choosing, measure the distance from the expansion cover at the back of the case to the drive cage. It looks like you might be limited in the length of card you can fit in there.
And to answer your initial question, the i5-3330 will not bottle-neck the HD 7870.