I am working on showing the shipping rates on the product pages, like in Amazon and eBay.
However, because I'll be calling the API a lot more then the usual cart page, I am wondering if there are any limits to USPS API that can prevent me from using the API on product pages?
Do you think I should use some kind of offline processing?
No - there are no general limits in USPS web-services specification. Although they have a right to limit you if there is a high-load from your side. Anyway if you will have limit problems they solved either by describing business need for that to USPS or by caching.
I think offline processing will be not an easy thing if you don't know how to do it obviously.
So - avoid offline processing as long as you can.