In the given example, yes, you can replace get started on with start.
I should start that sooner rather than
However, the two are not always interchangeable. Michael Owen Sartin wrote in a comment:
There is a slight difference between 'start' and 'start on.' One can start an engine, and the engine will be running. If one starts on an engine, the implication is that he is beginning a process such as painting or repairing the engine.
Regarding your sentence though, sooner rather than later is the usual expression, without the that in between.
Get started on (something) is an idiom. -- TFD
to begin doing something; to take the first steps to do something.
"When do we get started on this project?"
"I want to get started on this right away."