Is there an alternative replacement motor piece for 4.5 V Trains? Is it possible to find on a market a simple 4.5V electric motor, that can be installed into LEGO Part 564c01?

The service pack No. 5000 is more and more difficult to find.
First, check the assembly is correctly greased - they can throw a lot of sqeak out if they are needing a little grease even if the motor is fine! 

I have looked in to this myself, but the cost and trouble just isn't worth it - you can buy a used working unit for around £8 to £12 from a used / vintage Lego parts site. 

These motors are really good quality and almost last forever if looked after, so chances are you will get your £8 - £12 worth from a quality used one - regular off-the-shelf motors cost anywhere from £4 - £8 and that's if you can get a perfect size and voltage match not to mention getting the drive gear to fit it etc. Stick with buying a using original motor/full assembly and check it is suitably greased and looked after it

I have lots of Lego train sets and that was my answer to non-working motors. I look after them well, and they just keep going!