The shutdown has meant some federal staff, like air traffic controllers, have been working without pay.
Nancy Pelosi, Speaker of the House of Representatives, took to Twitter to blame the president for the disruption.
Hours later, Mr Trump announced a deal to fund the government and lift the shutdown temporarily.
Friday’s delays came one day after air industry unions issued a stark warning about the risk the shutdown was posing to public safety.
In total about 800,000 federal employees have been working without pay, or have been temporarily laid off, since funding to areas of the government halted 35 days ago.
These numbers include air traffic controllers, as well as US airport security screening staff.
The shutdown started in December, when President Trump refused to approve any new funding agreement that failed to include $5.7bn (£4.4bn) for his southern border wall.
Democrats in Congress refused to agree to those terms – so the two sides had been stuck at an impasse before Mr Trump’s most recent announcement.
How did the disruption come about?
The Federal Aviation Administration (FAA) halted arrivals for nearly an hour at New York’s LaGuardia Airport shortly before 10:00 (15:00 GMT).
It blamed air traffic staff shortages for the strategic closures – which also delayed flights at Newark Liberty International Airport and Philadelphia International Airport.
One passenger, named John Hitt, told the Reuters news agency he had to cancel a trip to visit his terminally ill aunt because of the delays.
LaGuardia airport said in a tweet on Friday afternoon that lengthy delays at the airport were ongoing.
There have been widespread reports about high absences among US airport security staff for a number of weeks.
On Thursday, the CEO of JetBlue Airways said the shutdown’s impact on carriers had so far been limited, but warned it was nearing a tipping point.
Southwest Airlines head Gary Kelly has described the shutdown as “maddening” – estimating they have lost out on $10-15m (£7.5-11m) in January sales.
On Friday, the Association of Flight Attendants issued a blistering statement in response to the delays.
“The aviation system depends on the safety professionals who make it run. They have been doing unbelievably heroic work even as they are betrayed by the government that employs them,” President Sara Nelson said in a statement.
“They are fatigued, worried, and distracted – but they won’t risk our safety.”
What is the latest with shutdown?
Mr Trump’s backing to re-open the government serves as a temporary solution but if no longer agreement is implemented, funding will lapse again on 15 February.
Speaking at the White House on Friday, he described federal workers affected by the shutdown as “incredible patriots”.
He also said they would receive the full back-pay they have missed.