William Kilpatrick posted the message to the social networking site as he finished work at Morrisons on April 24, 2011. Earlier that day the Old Firm had ended in a 0-0 draw at Ibrox.
The 22-year-old posted the message about Lennon, saying: "Get mair bullets and bombs sent boys, Glasgow Rangers Champions."
Stirling Sheriff Court was told that a week before the post was made it had been revealed that bombs and bullets had been sent to both Mr Lennon and high-profile Celtic supporter Paul McBride QC, who died last year.
After the posting was made a fellow supermarket colleague, Richard Heggie, reported it to police as he was "concerned that somebody might take it seriously and might follow up on this suggestion."
Kilpatrick claimed it was "just banter". However a jury found him guilty last month of causing a breach of the peace by posting a Facebook message that was of an offensive and threatening nature, thereby inciting others to commit violence, and placing the colleague who saw it in fear and alarm.
Sheriff Wyllie Robertson told Kilpatrick he would be "failing in his duty" if he did not jail him.
He said: "I have little difficulty in accepting that you did not appreciate the gravity of the matter when you posted those words. But you have been found guilty of a serious offence. Your comments towards Neil Lennon were offensive."