I got my 3rd Class Medical for a Private Pilot certificate in May 2013 at the age of 39. I'm now 41. Does the 60 month rule apply to my certificate or do I need to retest under the 24 month rule for over 40 year olds?
According to the table in 14 CFR 61.23 (d), the expiration date of your medical is based on your age "on the date of examination for your most recent medical certificate."
So, if you were 39 on the date of the examination, you're good for 60 months.
There is a possibility that the requirement to renew 3rd class medicals may go away completely soon. On Dec 15, 2015, the U.S. Senate passed the "Pilot's Bill of Rights 2". The bill would still need to be passed by the House and signed by the President before it would take effect. As of Dec 15, it had 152 co-sponsors of both parties in the House. According to AOPA:
Under the medical reforms of Pilot's Bill of Rights 2, most pilots who have held a valid third class medical, either regular or special issuance, within 10 years of the legislation’s enactment would never need to get another FAA medical exam. The rule would apply to pilots flying VFR or IFR in aircraft weighing up to 6,000 pounds and carrying up to five passengers at altitudes below 18,000 feet and speeds up to 250 knots.
Pilots who develop certain medical conditions, including a small list of specific cardiac, mental health, or neurological conditions, will have to get an FAA special issuance medical one time only, significantly reducing the time and money spent navigating the FAA’s medical bureaucracy.
For pilots who have not had a valid medical in the past 10 years and those who have never applied for and received a medical certificate, a one-time third class medical certification by an aviation medical examiner will be required. After a pilot has been medically certified once, either through the regular or special-issuance processes, he or she will also be able to fly indefinitely without needing to go through the FAA medical certification process again.