Nope. You cannot bring sick sim to hospital unfortunately. Besides most of the sickness didn't last long anyway so it was just minor inconvenience.
There are multiple ways to combat sickness :
1. Have your sim took a nap. It is especially helpful and free.
2. Have your sim drink orange juice. This is also free as long as you have access to fridge.
4. Have your sim drink tea. I guess any tea work but if your in doubt just drink healthy green.
5. Have your sim drink medicine. This one require you to have a computer to order the medicine. They cost $50 so this is theoretically the most expensive way to cure sims
6. Have your sim drink Snake Oil Serum. This one is gotten from a Level 4 Scientist that already know how to make it and STILL in the Science career. You need daisy in order to create Snake Oil Serum.