The only way to recruit Old Man Logan is to complete all Special Operations - Wastelands tasks.
15 Star Mastery of the Special Operation will NOT unlock the Hero.
One of the Tasks requires the defeat of an Epic Boss. This can be done by having the requirements needed to unlock the Epic Boss or by skipping the task by paying 25 Gold.
The requirements are:
Completing all threats.
Recruiting Iron Man, Spider-Man Noir, and Ant-Man
Finishing Deploys immediately as soon as they appear.
"We have just entered the Wastelands, where only Death remains. There are signs of some Villain Activity, so Wolverine will lead the strike team" — Mission Preview
Team-Up:Wolverine, Old Man Logan Location: Wastelands
Defeating a Mini-Boss will simply remove it from the full Boss battle. Minimum Score: (?) Team-Up:Old Man Logan (Forced) Wolverine (Optional) Boss Drop:
"Red Skull and Ultimate Ultron have teamed up and are invading all the domains in Battleworld. Old Man Logan will have to deal with this himself" — Mission Preview
Team-Up:Old Man Logan, Vision, Victor Mancha Location:New York