I take a duck bath when I get to work. Strip to the waist and splash cold water on head, shoulders and upper body. I wet the paper towels in the restroom (hell, they ain't mine so I splurge) and wipe myself down. I then use more paper towels to wipe myself dry. Brush my hair, apply deoderant and splash on a little aftershave. I bring a change of clothes and change in the restroom. Last step is clean up any water and mess you may have made.
Keep work shoes at work to make room in your backpack or pannier.
Ernest