To fit the six speed box you will also need the six speed's transfer box. You will also need the gear linkages, the five speeders will be too tight. The driveshafts are the same but the propshafts are a different length.
It is common for 2nd and 3rd gear synchros to fail on these cars.
A second hand five speed box will be just as old as yours and will most likely be just as worn, unless of course you are lucky.
Make sure you use a 75/90 GL4 based oil in the gearbox.
If you do find a suitable second hand g/box, check the condition of the output shaft. Please note that there are two sizes, 18 and 25 spline output shafts, unless you buy a box with the same as you currently have, you will also need the transfer box.
Personally, i would overhaul your current gearbox with new synchros and hub rings.