Pipe OD is the same for all schedules. 1-1/4" schedule 40 pipe has an ID of 1.38".
Depending on how tall the post is, a better choice may be fence post tubing that is nominally 1-5/8" OD.
And 1-1/4" sched 40 has an OD of 1.660", close enough. I do agree that the fence post may be a better choice in many situations, although when using threaded pipe it's easy to incorporate fine tuning for plumb. That can be handy in northern areas where seasonal ground frost can cause minor shifting.