Yes, it's a common issue, and no, there's no way to test everything. Really though, most functions/calls/methods/application items are called the same way.
It's really about testing the items that are important to you. What things are complex, or have failed, or might cause users to get upset. Test those things.
And don't try to test everything at once. A partial test plan is better than no test plan. Start building a test plan, even if it's one thing to look at. Then add to it as you learn more.
Talk to vendors. Many of them don't want to spend time testing new CUs, but if you pressure them, and lots of other customers do, they will test things. It should be fairly easy for them to regression test on a new CU. Fixing things that break might be hard, but few things should break. If tey can't test, perhaps they're not really testing their software.
This is certainly a hard problem, and a moving target. I think ultimately you, as the DBA, bear some responsibility to ensure your system works. If you can't test, then don't upgrade. However, be aware that if you don't bother to try to ever keep up, getting support from MS is hard. they'll want you to upgrade.