It's unlikely to take as long as an hour to be out and on your way at Changi airport. Taxis are no problem, also Uber and Grab (have a pickup location ready, e.g. Terminal 2 arrivals, door 3).
There is free WiFi at Changi, get a code by scanning your passport at a terminal or information desk, or receive by SMS if your phone can roam. Using this avoids having to buy an overpriced 'tourist' SIM from one of the network counters at the airport - if you can wait until the morning in town it will be cheaper and a longer validity package. Your best option on Orchard Road is the M1 Shop in the basement of Paragon (290 Orchard Rd) which opens daily at 11am. They will show you a range of options starting at S$5 with some calls and data for thirty days and $6 credit with which you can already buy a data pack. You will need your passport to buy.
There is now only 3G and 4G cellular service in Singapore, so a 2G handset won't work, and a dual-SIM phone that has only 2G connection on the second (non-data) SIM will have no service on that SIM.