T1, T2 and T3
Skytrain is free and links the three terminals in both the public and the transit areas. In the public area there are also link bridges: T2 to T3 on the Mezzanine Level and T1 to T2 and T3 through Jewel on Level 3. In the transit area, travellators link T1 and T3.
Skytrain hours are 5am to 2am on Changi's page, with a note that it closes earlier on some nights. Allow for walking late at night.
Jewel
Jewel connects directly to the T1 Arrival Hall on Level 1. From T2 and T3 it is 5 to 10 minutes over the link bridge. It is in the public area, so transit passengers must clear arrival immigration first.
T4
The public shuttle bus runs from T1/Jewel every 7 to 9 minutes and from T2/MRT every 5 minutes, 6am to midnight, and as a loop every 19 minutes from midnight to 6am. It takes about 10 minutes.
The transit shuttle runs all day, about every 8 minutes, and takes about 12 minutes from T2 or T3 and 19 from T1. You need a through-checked itinerary, and then security at the other end.
What it means for a hotel
Add the shuttle wait, the ride and security to any plan that involves T4. A stay at T3 with a 10 minute shuttle is a 30 to 45 minute trip to your gate.