2004 Jeep Liberty Power Steering Failing?

Could still be air in system. Could also really have been other parts such as the steering rack, the u-joint in the intermediate shaft, or binding ball joints. These may even have a screen in the reservoir that can get restricted with debris (not sure on this chrysler vehicle or just the FWD vans and such).

How long will it take to remove the air out of the system? And if it is the air wouldn't the Jeep been hard to steer to after it was fixed within that first day?

Most likely it would, but I am trying to give all possibilities. Have you checked the fluid level? My guess would be that it is the u-joint in the intermediate shaft binding up. The colder it is outside the harder it will typically be to steer.

Yeah they filled it up with fluid when they replaced the pump. Now is the u-joint generally an expensive part and how long can I go without getting a new one?

Usually is not very expensive. I would say you could go as long as you want. Its up to you really. Also remember to take it back to the shop if you are not pleased. If they told you this would correct the issue and it has not then you need to take up with them. Also my advice is not a final diagnosis, rather just informed advice as to what could be the issue.