Yes; however, going up is not helpful without also going sideways fast enough, and it’s the latter part where almost all of the propellant is spent.
The catch is that nobody can actually build a ladder strong enough. Space elevators from Earth require sci-fi materials.
You might even be able to take advantage of this by chilling the compressed air before it goes into the beanstalk, so that it absorbs heat from the surrounding air on the way up. Or have separate pipes with one carrying methane to burn as fuel. Lots of options to evaluate.
Yes, we've ruled out aluminum and wood. But don't give up! Got any single-crystal graphene? That might work!