Yes you do need the second "on"
Instruct Alex "on" = give Alex instructions show as well as tell.
Tell Alex what to do = change of verb you do not tell "on" what to do, you only communicate "what to do"
Tell and instruct both have an element of "telling" but instruct also has an element of showing/guiding/commanding somebody.
instruct Alex on is being more specific.
tell Alex what needs to be done = only a general outline.
"instruct Alex on what needs to be done, they need to ensure that A the orders go out in the correct order, B to receive signatures from the post office, C to hand the signature sheets into the account office every day, D lock up the shop every day, E to turn on the alarms before leaving and Finally F to ensure everything runs smoothly"
"Tell Alex to go the the post office every day get sheets signed look after the shop and lock up everyday"
Things change when there is a change of verb that does not show in dictionaries.