It's worth it. I would buy a used CPU. There is no reason to buy new on such an old system. CPU are extremely reliable. If it works when you buy it odds are it will work ten or fifteen years from now. Long after the system is obsolete or another component has failed.
Depending on the make and model of your motherboard. You may be able to get away with an Ivy Bridge or Sandy Bridge Xeon. I mention the Xeon as you can sometimes buy a used Xeon for a lot less money and get Core i7 performance.
For Example: This Xeon E3-1245 V2 should offer i7-3770 performance. If your motherboard supports it.
https://www.ebay.com/itm/SR0P9-684094-002-SM-INTEL-XEON-E3-1245-v2-3-4GHz-QUADCORE/172984234835?hash=item2846a9f353:g:~acAAOSwUYNaDcO1
An E3-1230 V2 should offer nearly identical performance for about $20 less. (No built in graphics on this model). An E3-1225 V2 has integrated graphics. At only 3.2Ghz but with hyperthreading. It should offer performance in most games somewhere between an i5-3570 and i7-3770.
Update: Looking at current used prices. The i7-3770 has fallen quite a bit. You'd be well advised just to buy an i7-3770 for $115 or the i5-3570 for $70. The i7 is worth the extra money. As you can likely get an extra year or two out of the computer. Than you would from the i5. No point in the i5-3550 as the used price is nearly identical to the 3570.