South Wales Police has arrested a 21-year-old man in connection with the incident at Kingsway just before 03:30 GMT.
Roads were closed for a time on Sunday morning to allow crime scenes officers to investigate.
The man's injuries were not life-threatening, said police.

A trailer was taken from Burton's Foods Ltd on Ty Coch Way, Cwmbran, Torfaen on 17 June.
The two men, from Liverpool and Cheshire, had been charged with theft.
A hearing at Newport Magistrates' Court heard a third man will appear in court in relation to the alleged theft on Friday.

The device was stolen during the band's concert at Sunderland's Stadium of Light on 25 May.
Northumbria Police found the phones at a house in Whitley Bay hours later after using a GPS locator app to find it.
A 25-year-old woman has been arrested on suspicion of theft.
Det Insp Dave Swinburn said the force advised concert goers to be "extra vigilant" and recommended smartphone owners install tracking software on their devices.

Healthcare Improvement Scotland said several senior charge nurses and clinical service managers across three wards raised issues during an inspection visit.
NHS Grampian was asked to supply an improvement action plan.
Senior inspector Ian Smith said the action plan had given "assurance".
He added: "We will continue to follow this up with NHS Grampian on future inspections."
The inspection visit took place in April, as part of Healthcare Improvement Scotland's programme of inspections on the standard of care for older people in acute care in Scotland.
It was to check that older people were being "treated with compassion, dignity and respect".
NHS Grampian was said to be "performing well" in a number of areas.
An NHS Grampian spokesperson said: "We welcome this positive report which highlights the many examples of good practice in the care of older people in hospital, particularly the compassion, dignity and respect with which our patients are treated.
"We are also pleased that 93% of the patients surveyed said that the quality of care they received was good, reflecting the warm and caring interaction observed by the inspectors between staff and patients.
"The areas for improvement identified in the report are already being tackled, and an action plan has been agreed.
"Although acknowledging that patient care was not compromised, the inspectors asked that staffing levels in three wards be addressed."
A 22-year-old man has been taken to hospital following a "serious assault" in Swansea city centre.
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