A good way to verify if the update has no problems?
As TheGameiswar has mentioned, the upgrade adviser is incredibly helpful for pointing out issues that can arise with the upgrade. However, it will only check for things IN the database. So if you have ad-hoc queries being ran, or worse yet sql files scattered all over the place, those will not be tested.

When I upgraded our SQL 2000 servers to 2014, we had ad-hoc queries and sql files all over the place that were being executed and we needed to test to make sure they would not break as best we could. So what I ended up doing was copying our production databases and upgrade them to 2014. After that, I started doing replay traces on production for a week to capture all queries being ran. This allowed me to replay those traces on the 2014 servers to see if there would be any issues. It helped me find a number of issues that needed to be fixed. 

One very important thing to note when doing replay traces is you need to make sure the database IDs match between the production and dev environments as that's how the traces know what database to run each query on.