I've been in/out of Manila at least a dozen times so will weigh in and give a few tips and tricks. The first is to note that "Manila" is actually the name of a collection of over a dozen separate cities collectively known as Metro Manila, of which "Manila" is one. The airport is located in Pasay City, in the south west area. I assume you are spending the two nights in Manila on your way to PG to rest up and time adjust from your flight from the United States. A couple things to keep in mind as it's your first trip to Manila/Philippines. (I don't know if you've been to other SE Asia destinations before.) Manila is a crowded, hot and humid, traffic choked, heaving hive of humanity. It can also be quite sketchy to downright dangerous...with many streets and cities you don't want to be on or about after sunset (and many even during the day). Don't get me wrong, the Filipinos are some of the most warm hearted people you will meet, but the extreme poverty is endemic (and will break your heart the first time you are there) and this breeds both petty and serious crime. (Hence the shotgun toting security guards at every McDonald's.)
Okay now that I've scarred you out of your trip, where to stay. I second the prior reccos for the Belmont Hotel, or any of the others within the Newport Mall area, directly adjacent to NAIA (Hilton, Newport, Marriott, Holiday Inn Express). These will all be clean and comfortable and well within your budget. You want to be near the airport because of the horrendous traffic day and night. Why waste an hour or two on arrival getting to a hotel across town. You may also want to spend the night there on your way home, and again avoid the traffic stress getting to the airport to make your flight. There is also the Newport Mall itself for all your eating (Cafe Mary Grace recommended) and shopping needs in air-conditioned comfort. My other suggestion, just a short drive from the airport, if you want something a little more lively, is the Hyatt City of Dreams Hotel & Casino. A super luxe property which you should be able to snag within your budget, which gives you the option taking your chance at the tables and many delicious food outlets. The Mall Of Asia, one of Asia's largest, is a 5 minute ride away for any shopping.