Jeep cold engine noise

That engine is known for having problems with the hydraulic valve lifters as it ages.
If I recall correctly, there are replacement parts of an updated design, in order to overcome this engine’s inherent valve problems.

However, only you know whether you are willing to spend some $$ on this 13 year old vehicle. If you are planning on driving it for 2 more years, I would advise spending the money–as long as everything else on the vehicle is in good shape. The good news is that the cylinder head does not have to be removed in order to replace the lifter mechanisms, so the job won’t be as expensive at it might be on some other models.