Yes, you can often take an old, dirty press and clean it up to give it a new life. There shouldn’t be any problems taking a press apart to clean it. I wouldn’t recommend painting any of the parts especially in or on moving parts areas. Clean it very well and make sure there is lubricant on all moving parts and bolts. You can use white lithium grease for moving parts and threaded components. Use WD-40 or light machine oil for bolts and screws that you can’t get lithium grease on.
Don’t take the print heads completely apart unless you need make repairs or adjustments. They should be able to be removed intact and you can put the entire head in a tub of paint thinner to clean it. Do not use acetone. Just regular mineral spirits/paint thinner. Then soak it with WD-40/light machine oil and re-lube it well. Be careful of any plastic or nylon parts. Make sure any lubricant used is not aggressive on all non metal parts.
If you are worried about how the press goes back together and if it will affect registration, then make sure you put it back the same way it was. Most of the registration components are a part of the print head. However, if you mix up the parts and re-assemble pallet arms and such in different positions, it may be possible that you could experience at the very least operational problems. It is best to keep all the parts in the original positions. You can make notes on masking tape to label parts for re-assembly. Just remember not to lose them during cleaning and lubrication.
Basically, a good cleaning will help any press. If your print head is broken, there may be replacement parts available on the internet. If you have a popular press, it may be possible to purchase a second nonfunctional unit to use for parts. There are many resources on the internet. All it takes is a little research.
Thanks for reading!