A few things to take into account:
With Grab: From NAIA -T1 to Metro Manila (Ermita area/Roxas Blvd.), I paid 276 PHP = approx. USD 6.
With Uber: From Metro Manila (Ermita area/Roxas Blvd.) to NAIA -T3, I paid 178 PHP = approx. USD 4.
If transferring within Terminals:
When departing from T1:
Places to visit, eat and/or stay:
For sunset views of Manila, head out to Sky Deck at the Bayleaf Intramuros Hotel which offers a good selection of Filipino food as well as a 360 degree view of the cityscape.
Once there, head out to their food courts and try out Gerry's Grill and The Aristocrat Restaurant.