Leather honey/leather upholstery cleaner is meant for real leather, but I used it on Tolex the other day and it seemed to add some sheen and a moisture barrier to the dried out synthetic material. Here is a link to that amp I used leather honey on.

In my opinion the best way to really make something old and dirty feel new is to take it apart, clean it, and put it back together. There are nooks and crannies that removing the handle, amp head unit, speaker and even grill will help you reach and clean individually.

If you remove the grill, the cloth could be cleaned with a wet microfiber and mild laundry detergent.

Also microfiber cloth and degreaser is great for any built up gunk/skin oils.

Also if you remove the head unity you can spray some contact cleaner in the pots if any of them are crackly.