Are there natural treatment/treatments for arthritis that also reduce cholesterol?

Maybe. These natural treatments include weight loss, exercise. We can not discount these as physical therapy is the cornerstone of arthritis therapy. Most of the medications used for arthritis do not effect the cholesterol, although sometimes Folic Acid is added to help protect the side effects of Methotrexate and this can lower hyperhomocysteinemia which can be related to low folic acid.
Yes,several! 3-omega fatty acids found in fish or flaxseed oil will work in 4 capsule a doses. Sustained release Niacin works. Both are available in rx strength. Exercise will increase hdl, potentiate weight loss . Finally, modest alcohol consumption works, but in moderation.
Yoga & Ayurveda. Natural treatments in conjuction with standard treatment if one is interested may consider yoga, diet, exercise, weight loss, good sleep hygine, meditation, accupressure . Alternative medicine like ayurveda herbs based on body type (prakuruti) and toxins/energy imbalanc (vikruti) for which one may see ayurveda practioner as they are different herbs one to use because it is indivulized therapy.
Not effectively. A good remedy for arthritis like glycosaminoglycans has little benefit in lowering cholesterol. An over the counter treatment like red yeast rice extract or high potency fish oil will not help arthritis much but does improve your cholesterol profile although probably not as much as prescription drugs like statins. Also, such treatments are not well regulated and can contain contaminants.