you can do it

used to commute to a place w no shower just a private bathroom
shower in morning before work & apply good deodorant like Tom's unscented
on arrival strip all clothes head to foot & bag them
use a washcloth as if you were in a shower
a liquid soap diluted is easier to use & rinse off
for hair use diluted clarifying shampoo in a plastic cup, put your head down & do the deed, then use the cup to rinse
use a good hand towel (that you bring) to dry off
apply deodorant, Tom's unscented is good
dress in new dry clothes head to foot
if you are still damp "down there" you can stuff paper towels in your shorts at your desk for 10 min

it helps to arrive early & get it all done before the rest of the gang shows up

good luck!

found a cpl older threads about the same thing

Commuting with no shower at work?

Must Wear Suit / No Showers - How Can I Bike To Work?

How to clean yourself without a shower?

Riding a bike to work and being sweaty and smelly?

Cleaning up after riding to work?

What do you use to freshen up after your ride?