Ferries (for both cars and passengers) cross to Udo's western port in Haumokdong (하우목동항), or sometimes southern port in Cheonjin (천진항), from Seongsan port in eastern Jeju. The crossing takes 15 minutes and ferries leave hourly. Bring your passport.
You need to fill in a boarding pass before purchase and show photo ID, so arrive at least 30 minutes before departure. Hand in the boarding pass on boarding, and fill in another one when returning from Udo.
Ferries don't operate in bad weather.
At the time of writing, a ban on bringing rental cars to the island was in place until August 2018, with uncertainty about whether this would be extended. Check before bringing a rental car to Udo.