The capabilities of smartphones vary widely. Some smartphones cannot synchronize tasks, or they restrict the number of task categories. See above for recommendations on evaluating or choosing your mobile device.
You can do TRO without a syncing smartphone, no problem. If you ever work away from your desk, print out your Do Today list, make notes on it, and process the changes each day. Some people love this. Look for a task list that has a tolerable print format. It doesn’t need to be fancy.
You can also use your mobile calendar to see a mobile Do Today list away from the office.