Time is used in many different ways as a goal metric. For example, a player may be informed that he has a certain amount of time to complete a task. This is frequently presented in a dramatic narrative context such as time before a bomb detonates, or before a ship sinks. Other contexts, such as those seen in sports where the game has a specified duration, are much more arbitrary.
Another use of time as a goal metric is to use it as a measurement against which to directly compare (and determine) success. For example, in most racing games, the winner is determined on the basis of who traversed the race track in the least amount of time. In fact, some racing games penalize competitors that perform bad maneuvers (like crashing obstacles) by adding seconds or minutes to their final time.
While it is possible to keep track of the time it takes to complete a puzzle yourself, in order to have your own 'best time' and try to beat it, there are no real restrictions on (or rules about) time in the game sudoku.