Mr Cole, 42, was returning to the UK on a flight from Dublin when the incident happened. Police were called to Manchester Airport following reports that two passengers were drunk and would not co-operate with Aer Lingus cabin crew.
Lee Patrick Byrne, 28, from Dublin, and Gregory Horan, 26, of no fixed abode, were arrested on landing.
Byrne pleaded guilty at Trafford Magistrates' Court to a racially aggravated public order offence and being drunk on an aircraft, the Crown Prosecution Service confirmed. Horan also admitted being drunk on an aircraft.
The defendants were bailed to appear at Manchester Minshull Street Crown Court for sentencing on January 20.
Both men could be imprisoned for the offences with a maximum five-year term for being drunk on an aircraft.