Not "start to do something", but yes definitely "have the opportunity to do somthing". It also has the meaning of doing something after a certain period/length of time or after an interruption or wait. Eg. After all these years I finally got to visit Rome. After tidying up, washing the dishes, then putting the kids to bed, I got to sit down and watch TV.

Here's a few examples: You keep talking about how cool it is for someone meet their favourite celebrity, but did you actually get to do it yourself yet? Did you get (a chance) to play on your new computer yet? I didn't get (round) to doing the housework this morning, but hopefully I'll find time for it tomorrow.