i would (even against better judgement) get a second opinion on the shift linkage, or even just replace some of the pieces, the whole pivot arm/rods in the engine compartment is $55 and would take about an hour to swap in. @ $150 is the same price as a true diagnosis of whether the transmission is broken or not. AND that part of the shifter is sloppy when new and easily worn or breakable.
the 'real" diagnosis: does it grind, give resistance, or not go into the spot 5th is? if it shifts into gear fine, but grind only when you let out the clutch, then it could be the trans. can you put in in 5th gear when it is not running? if so try parking facing downhill on a step hill and put it in fifth and let the clutch out, what happens?