I have a 2008 Mac Pro and it has no wifi card in it. I have a time capsule. Can I connect the time capsule to my router via wifi and use an Ethernet cable from the time capsule to my Mac Pro to access the Internet?
This depends on the type of WIFI network that is being produced by your WIFI router, as to whether the TC can extend it or not. Basically, the best way to find out is to experiment on your TC - fire up your Airport Utility, and in the Wireless configuration option, look to see if it will let you extend rather than create a wireless network, and then see if will find your existing network.
EDIT: I should add that the brand of wireless router is a primary consideration as well. Many brands don't play well with TC, AE, or any other Apple networking gear.
I tried to clarify your topology in comments above. I presume you want a wireless bridge between your router and the TC. The answer to that question, as stuffe suggests above, is to try it. However, I would suggest using the wireless router to control your traffic, and putting the TC in bridge mode. If possible, I would connect the TC to the router via Ethernet cable, even if the wireless bridge is possible.