Hour-long layover with Philippine Airlines at MNL?
 

I'm trying to purchase a flight ticket from Taipei to Singapore, but most of the flights have sold out except for one that involves Philippines Airlines stopping for an hour at MNL airport in Manila

Would that be enough time to pass through customs/security/etc to make the flight?

I believe you will clear customs in Singapore, not Manila. If your flight is not late then 1 hour should be enough time. Try to sit as far forward on the plane so to minimize your time in getting off the aircraft. I always allow at least an hour and a half but if that is not an option then go for it. If the flight from Taipei is not very long then the chance for a major delay is small.

It's pretty hard to give advice when you don't specify the airline or airlines involved. Typically, if the itinerary is on one airline, or two airlines with reciprocity, you would not go through entry procedures at the midway stop since you do not enter the country. You might even not change planes. If you cannot check your luggage all the way through, then you probably would go through entry procedures before checking in to the second flight, really squeezing a one-hour stop. On any transfer, some security for boarding the second flight is to be expected. All this is guesswork.
Another guess: Why not fly non-stop, and maybe cheaper, on Jetstar Asia?