How can I clean surface rust from damascus or twist barrels without removing the pattern?

An old trade process for polishing metal used white black-board chalk sticks in combination with oil. First coat the barrels with a light film of gun oil or KleenBore Formula 3 Gun Conditioner. Then rub the barrels with the chalk stick, focusing on the rusted areas, but to include the entire surface. The rust is lifted by the chalk sticks without damage to the damascus pattern. Wipe clean with a soft rag and repeat as required until all the rust is removed.
After cleaning all residual from the barrels with G-96 Brand Triple Action Gun Treatment or Ballistol, consider a coat of Flitz Rifle & Gun Wax or Renaissance Wax Polish.

Nu Finish Car Polish and Birchwood Casey Stock  Sheen & Conditioner will clean, add lustre, enhance the pattern contrast, and protect damascus barrels.

Courtesy of Joe Wood for restoring and protecting damascus barrels after cleaning:

Formby's Tung Oil is by far the best metal finish I've used. It is very easy to apply (or remove), dries quickly, and seems to have "miracle" restorative properties when applied to faded damascus barrels.

Place a long wooden broom handle in the breech of one tube and tighten the handle in a vice so the barrels are horizontal. Wet a 4 inch square cloth patch with the oil and wipe it on, moving quickly. It might help to rotate the tubes using another wooden dowel in the muzzle of the other barrel. One patch should do both barrels. Let dry for 3 to 4 days then admire your work. The Tung Oil also can be used to protect case colors on the receiver."