Medicare Adult Day Services Act of 2013 - Amends title XVIII (Medicare) of the Social Security Act to: (1) cover certified adult day services furnished in a certified adult day services center and meeting specified requirements; and (2) require the Secretary of Health and Human Services (HHS) to increase by 3% the payment amount otherwise effective for such services furnished in a rural area on or after January 1, 2014, and before January 1, 2016.