Either OEM or Federal Mogul will do.

You can try Minerva Tire Gallery (Binan or Alabang), they have decent mechanics (and the manager of the Binan branch is a friend), but they don't really have a complete set of spares for the Lynx.

You can Do-It-Yourself at home for the initial check-up... take off the radiator cap and check that the seals and spring are still good... but heck, it's so old you might as well replace it, anyway. Once you've replaced the cap, take it to a local shop and have the coolant drained, clean the radiator, then put in new coolant and water (50:50 coolant:distilled should be just right)... drive it into traffic and observe. If it stalls but doesn't overheat, then you probably have to replace one of the two electronic items I mentioned above. If it overheats or starts getting hot, check your water pump and radiator hoses for leaks. There are sometimes pinpoint leaks in the radiator hoses that only show up when there's water going through them at high pressure.