Will eukaryotic RNA fold in the same way in prokaryotes?
As you have mentioned ions and temperature affect RNA structure. There are also different types of RNA structures and their dependence on ions are different. Mg2+, as Mad Scientist mentioned, stabilizes duplexes; so do monovalent cations like K+ and Na+. However, Mg2+ favors duplex over quadruplex if the same RNA can adopt both these conformations.
Dependence on temperature is a trivial case.

Ions and temperature should be more or less same for prokaryotes and eukaryotes unless we are talking about extremophiles. 

Apart from these factors I can think of two other factors that can cause difference in RNA structure between prokaryotes and eukaryotes:


Osmolytes
RNA binding proteins/chaperones (Already mentioned in comments by Mad Scientist)


Osmolytes

It has been shown that TMAO (Trimethylamine N-oxide) stabilizes RNA secondary structures. The metabolism of TMAO is different in prokaryotes and eukaryotes. 

From this paper:


  Although eukaryototes can endogenously produce L-carnitine, only
  prokaryotic organisms can catabolize L-carnitine11. A role for
  intestinal microbiota in TMAO production from dietary carnitine was
  first suggested by studies in rats; moreover, while TMAO production
  from alternative dietary trimethylamines has been suggested in humans,
  a role for microbiota in production of TMAO from dietary L-carnitine
  in humans had not yet been demonstrated30-32. The present studies
  reveal an obligatory role of gut microbiota in the production of TMAO
  from ingested L-carnitine in humans (Fig. 6c)


I cannot ascertain that this will affect RNA folding but is possible.

RBP

This is something that you can be certain about. Some RNAs require protein counterparts to adopt a functionally capable structure. In the absence of the protein they may not form the relevant structure. So if an RNA needs Hfq then you have to express it in the eukaryotic system where you want to use the RNA (and converse).