Stick on horse to finish first in old Wild West show (5)
I believe the answer is:
rodeo
'wild west show' is the definition.
(I've seen this before)
'stick on horse to finish first in old' is the wordplay.
'stick' becomes 'rod' (I've seen this before).
'on' says to put letters next to each other.
'to finish' indicates one should take the final letters.
'first in' says to take the initial letters.
The final letter of 'horse' is 'e'.
The first letter of 'old' is 'o'.
'rod'+'e'+'o'='RODEO'
(Other definitions for rodeo that I've seen before include "Wild West contest" , "Annually" , "stock venue?" , "One should be up for this event" , "competition in cowboy skills" .)