Well you can give it a try - note that the module is only for Ubercart 1.x. I was under the assumption that UPS used cURL to begin with, but perhaps not. And to be clear cURL is something that needs to be installed on your server - so, if your version of PHP was compiled without it, that UPS-Curl module won't help. If you try it and it still doesn't work, you need to contact your hosting company and have them look for any error logs, and possibly install cURL for you.

(By the way, did you see any error messages in Watchdog? Do you have access to the server's error logs as well? Looking at that might help you solve the problem.)