The 28-year-old, who cannot be identified, was arrested in west Belfast after a taxi driver raised the alarm.
He hurled abuse at officers, prosecutors said.
The man admitted counts of disorderly behaviour and being drunk in charge of a child in public.
Belfast Magistrates' Court heard his son was distressed and outside without shoes when police spotted the pair in the Suffolk Road area last August.
Arrangements were made for the boy to be taken to a relative. At that stage his father became abusive and aggressive as police arrested him.
Defence barrister Declan Quinn told the court his client had only been drinking due to being the victim of a mistaken identity in a previous serious but undisclosed incident.
His actions have resulted in him suffering the "opprobrium" of his family, the lawyer said.
Mr Quinn added: "He is genuinely remorseful and disgusted with his own behaviour, and concerned about the impact on his son."
Based on recommendations in a pre-sentence report, District Judge Harry McKibbin imposed six months probation for each offence.
The judge also banned reporting of the defendant's name to ensure the victim cannot be identified.