To do Euclidean geometry, we only need a compass, protractor and a scale (maybe a set square to make our life easy) to make all possible geometrical constructions. Suppose a geometer living on a curved space wanted to do geometry similar to how we do it on the flat plane, what would be the tools he would have in his box?
Let's suppose he is primitive and still doesn't have any computer /3-d plotting methods (otherwise we'd he a trivial answer for this question)
What exactly do you mean by “do Euclidean geometry”?
The classical theorems of Euclidean geometry would be compass and straightedge only, and many of them could be performed in a curved world just as well. In fact I seem to recall that Euclid's Elements delays the use of the parallel postulate as far as possible. Of course those theorems that don't need parallels could also be called neutral geometry instead of Euclidean geometry. And the curved counterparts to actual Euclidean theorems might be harder to construct, where they exist at all. If you want to do computations there, you'd probably have strong inventive to develop a lot of tools that we here haven't needed until only a few centuries ago. Calculus and differential geometry prominently among them.
In everyday life we probably want to add tools to measure distances and angles. Where in Euclidean world you need both of these, in a curved world in theory one of them might be enough, since you can e.g. define lengths in terms of angles there. But the translation involved goes via trigonometric and hyperbolic functions. So I'd expect they would still have distance and angle measure tools for convenience. And probably some slide rule like affair to compute those trigonometric functions. Possibly accompanied by precomputed tables for trigonometric functions, or perhaps a pocket calculator if that level of technology makes sense for your question.
Why do you do geometry? I've heard that one of the earliest applications was to measure and describe the shape and size of patches of land. The term geometry actually translates to “earth measurement”. In our (only very slightly curved) world, we tend to use drawings at scale for this purpose, namely to create land allocation maps. In a (more) curved world, drawing at a scale is a less trivial concept: you can't just scale things down without changing their shape. However, for beings in a curved 3d space there should be a way to work around that by creating an even more strongly curved drawing surface for the scale drawing. So I'd expect applied geometry there would involve drawing “paper” at a set of well established constant curvatures. A bit like the actually spherical globe you might have sitting on a shelf to provide a scale model of our curved earth. Even outside applied situations, constructions on a more strongly curved surface might make properties coming from the curvature more pronounced and thus easier to explain and work with. I'd expect students in school to use the same kind of curved drawing surfaces for that reason.
While they are at it, they might continue to also work with drawing surfaces of zero or opposite Gaussian curvature. See Studying Euclidean geometry using hyperbolic criteria for a discussion of hyperbolic world inhabitants studying Euclidean geometry there.
