Can I access Creature Slaughterdome without owning the DLC?

The Creature Slaughterdome is a downloadable content "Circle of Slaughter" map for Borderlands 2.

• Access can be gained for players who do not own this DLC by accepting the mission chain Creature Slaughter with a player who owns it. Access is granted until the mission chain is completed.

http://borderlands.wikia.com/wiki/Add-on_content

Entering without it is still possible if a pre-order or paid owner's game is joined. If the mission is obtained but not completed this way, access is granted independently until mission completion.

http://borderlands.wikia.com/wiki/Natural_Selection_Annex

How do I accept the mission chain Creature Slaughter? If we try to travel to the DLC area, would I prevent them? Or do we travel normally and I pick up the mission chain upon entering?

With your friends in your match, provided you as the owner have the Creature Slaughterdome, you will be able to enter and start the quest chain.

The quest chain is begun by interacting with the NPC in the Annex, which will activate it for your entire party.