You may actually experience some bottleneck to be honest. The G620 is a dual core without HT or Turbo enabled. This means you have only two cores that will run at ~2.6GHz only. Are you currently using the integrated graphics, or do you have a dedicated GPU? My recommendation would be to just be patient for a month and buy both the CPU and GPU at the same time so that you don't put yourself through unneeded frustration, and you only have to open the case and upgrade in one sitting, which equals less down time.
I need to correct myself here...above, I mentioned an i3 3470...which was a typo...The 3470 is an i5. A third gen i3 may run ok with the 460. The i3 3250 is dual core, but it has HT, unlike your G620. It does not have Turbo enabled, but it is set at 3.5GHz, which is quite a bit faster than the G620 as well. Not a bad clock speed with 4 logical cores in operation, and would be a good budget choice.
Here's a link for the stats of the i3 3250:http://ark.intel.com/products/74744/Intel-Core-i3-3250-Processor-3M-Cache-3_50-GHz
There are many types and flavors of new games out there, so it really depends on the titles you are wanting to play. Some games are really CPU heavy, where others are optimized for more focus on the GPU, whereas others may be affected by your RAM. It all depends on the engine and the developers, and the choices they have made in their coding. No two new titles will tax your system identically.
If you are wanting to push a bit more life out of your mobo and system with these upgrades, you may want to look at the third gen i5s. They have 4 physical cores compared to the logical ones of the i3. They don't have HT, but they do have Turbo enabled. The i5 3470 is a 4 core with Turbo. It has a base clock of 3.2GHz and Turbos up to 3.6GHz. Depending on the motherboard and chip, you may be able to push that a little more with some overclocking. Four cores at 3.6GHz+ should sit you just fine with the RX 460 for a good spell. Again, it really comes down to the titles you are looking to play and how well they are optimized.
Here is the link for the specs on the i5 3470:
http://ark.intel.com/products/68316/Intel-Core-i5-3470-Processor-6M-Cache-up-to-3_60-GHz
Here is a link to ebay to show you what they can be picked up for:
http://www.ebay.com/itm/like/302164567173?lpid=82&chn=ps&ul_noapp=true
I would say to check out the i5 3570, but I am not sure what your budget is, and they are still rather pricey...But maybe you should give them a scope anyhow.
I really hope this long winded information is helpful!