I've been cycling commute for almost 30yr now, some places shower and others no.
Most important is to have towel and spare work cloths, wash in the toilet and off you go...
I think as long as you are putting on a change of shirt, shorts and socks when you get there, a wipe around the dark corners is quite enough.
People get overly paranoid about personal hygiene it seems to me.
there is this stuff + baby wipes..
When I was riding to work our place got flooded and we lost use of it including all the showers - new temporary place didn't have any but did have sinks and I bought a push on shower hose that fitted over the taps. Surprisingly effective, you could lean right over the basin and get your top half well and truly rinsed without making too much mess on the floor. Car to work once a week or so and clean shirts etc came in then.
Have commuted by bike since I started work proper in '92 with a variety of cleaning options. Some sound advice here. Leave a well stocked wash bag there with talcum powder. Those quick dry towels are great and they're light. Plenty of socks, pants, ... and 2 pairs of shoes (one black , one brown) can be left there if you have a space. One tip in Summer if you get really hot - leave ten mins early and walk the last half mile. You will cool off considerably as your heart rate reduces and be in a better state. Panniers are excellent and reduce back sweat and back pain. I cover 120-150 KM a week now and find it has made a huge difference to my weekend off roading and my expanding mid 40s gut.
Always make sure you have spare undies and socks and keep track of your washbag, nothing worse than getting to work and realising you're out of deodorant!