Inspired by the similar question regarding a torus, imagine that you're a flatlander walking in your world. How could you distinguish between your world being a sphere versus being a projective plane?
It seems like this would be somewhat harder because you can't necessarily use an argument about non-positive curvature, and you can't (easily?) take advantage of the fact that the projective plane is non-orientable and not embeddedable in $\mathbb{R}^3$.
I would also be interested in any methods that could be used in this case, would wouldn't tell you anything substantial if you were on a torus.
Since the projective plane is $\mathbb{Z}_2$ quotient of the sphere, it is impossible to distinguish the two if you confine yourself to a simply-connected region.
If you don't, then you can take two identical, oriented objects, leave one where you are, and walk with the other one straight until you get back to the first object. On a sphere you will find that the objects still look identical, and on the projective plane you will find that the objects are now oriented differently.