Off means away from the place in question to a distance, so you were off and running meaning you had made a start and doing a good initial progress. Pull on refers to holding onto and move something towards oneself e.g she pull on the rope with all her might.

‘Off and running’ means that you’ve started and the project is now active. It’s used positively. It seemed like it took forever for us to open this restaurant. I’m so happy we’re finally off and running. ‘on’ isn’t strictly necessary here, but it’s more natural. ‘pull on’ is less constant than ‘pull’. For example, starting a lawn mower would typically require one fast and hard pull. You’ll be a better dancer if you pull your partner closer to you. (Not ‘pull on’) Pull on this cord to let the bus driver know that you want to get off at the next stop. (‘on’ isn’t necessary but it indicates that you’d just have to do it once to let him know.)