Will this count as an upgrade?

I have the iPhone 4, and my aunt is gonna give me her old 4S. I'm under contract with Verizon until April 2015, but I want to use the 4S instead of my 4 because mine is acting up. If I take the 4S to Verizon, can they activate it with my phone number without it being an upgrade? I can't afford the upgrade fees but I really want the 4S

No you can use the iphone 4s if your Aunt was using verizon if she wasn't you just have to unlock it

No. The iPhone has a contract till April for particularly THAT ****iPhone 4**** and not 4S.
You'll have to use your iPhone 4 till April 2015 as the number and SIM is fixed up with the 4 and it can't changed until the contract is completed(i.e. Unlocked in technical language)

Well, if you are really desperate for an iPhone 4S, then you should go to a 3rd party unofficial repairer of iPhone and other stuff so that he/she can help you out get your iPhone's SIM unlocked.