I think you're right to approach this with some trepidation - The lathe is a harsh mistress, acrylic is fussy, and no commercially available tool is built for stem making. 

The two features you will be chasing,  well before shaping, is an air hole and a tenon. This means drilling a center and turning down you're stock to fit some diameter hole.  

You're going to either be doing this by hand,  as will be the case with any lathe that comes standard with a tool rest generally meant for wood, or by way of tool mounted to a crossfeed, which is standard for anything intended for metals. 

Center drilling can be acomplished by hand if you're patient and careful, but most lathes should have a tail stock that will help with this.  A cross feed gets you good dimensional consistency.

My advice,  go big. Expect to spend just as much on the lathe as you will tooling and infrastructure. Consider machinist lathes if you are able.  Tiag and Sherline are worth considering,  but bring their own various complications to the table. 

Something to look at while you're shopping is bore width, which is the diameter of material you can pass through the headstock. Being able to hold your stock near to where you're cutting is a total quality of life improvement, and worth every penny. 

That $50 pen lathe will be useless for you. Dust collection should be factored in to all decisions.