"Pulling an oar" figuratively means putting effort into a group enterprise. Think of competitive rowing with several rowers propelling a boat in a rhythmical, synchronised fashion. But there is a grammatical issue with the expression the way you have it. "Our" needs to agree with "everyone", rather than with "of us". So it should be written as "Every one [or simply "each"] of us must pull his or her oar!”, or, using "singular they", as "Each of us must pull their oar!”.

Everyone has to do their part, especially if it is a team effort. If it is not, it would mean you have to do your job or task, and just get on with it even if you are not happy doing it.

Everybody has to contribute to the goal together and equally. This is a metaphor for working hard. The person is in a boat and is making a contribution by pulling their oar so that the boat moves. Similar sayings: Everyone has to pull their own weight. Everyone has to work together. Everyone has to work as a team. No room for slackers. Together we conquer, divided we fall. Teamwork everyone! It could also mean that you have to care of yourself because no one else will. Depends on the context. Good luck