True. Could be the ATF. When you said it wont engage on the early morning because the tranny is low on fluid or perhaps the filter is almost stopped up and the pump cant pick up fluid. It would be impossible to guess what is the real problem or cost would be, except it does like transmission.
Change the fluid and the filter on that tranny. This will be the cheapest remedy before shaving too much money for rebuilt. Up date what happen next.
When you say "pag mainit na makina" is it because you've been running the car in drive or has it been on idle in park the whole time? When you keep the car running in park the engine gets warm but not the transmission because it needs the car to be moving for the fluid to flow and get warm. Check the links it may be that it shows you are on "R" but you are not actually on it because it is not aligned. If you are lucky it's just this problem and nothing else. But from the sound of it you will be needing a rebuild soon. When my BMW was rebuilt it was because I lost reverse completely. See if you can buy the rebuild kit and get a good A/T tranny shop to fix it for you.