On the up, exist for a long time (3)
I believe the answer is:
era
'long time' is the definition.
(a long period of time)
'on the up exist' is the wordplay.
'on the up' says the letters should be written backwards (in a down clue, letters go up).
'exist' becomes 'are' ('they exist' means 'they are').
'are' backwards is 'ERA'.
'for a' acts as a link.
(Other definitions for era that I've seen before include "Period, age" , "Long period of history" , "Specific time" , "Epoch, period" , "Important date" .)