Consultations are being held between Yemeni political factions in Saudi Arabia's capital Riyadh in a bid to revive UN-led peace talks aimed at ending the war.
“We immediately announce the release of the first two fuel ships through the port of Hodeidah,” Mr Mubarak said on Twitter.
Mr bin Mubarak said he had also been instructed to “facilitate all the arrangements for the release of all prisoners” as well as “opening Sanaa airport” and “opening roads in the besieged [city of] Taez” in the south.
The ports of Hodeidah and Salif are controlled by the Iran-backed Houthi rebels. The Saudi-led coalition has said the Houthis use them as launching points for military and marine operations.
However, some areas on the outskirts are under the control of pro-government forces.
A peace deal was signed by Yemen's warring sides in Stockholm in 2018 which aimed to keep the ports operational.
Air and sea access to Houthi-held areas are controlled by the coalition, which intervened at the request of the internationally recognised government in 2015, after they it ousted by the rebels.
Mr bin Mubarak said he is “affirming the government's firm position in supporting any efforts that alleviate the humanitarian suffering of our people in light with the positive atmosphere the Yemeni talks in Riyadh.”
“I received clear directives from President Hadi to take necessary steps to facilitate all the arrangements for the release of all prisoners, opening Sanaa airport, releasing oil ships via Hodeidah, opening roads in the besieged Taez, to alleviate the suffering caused by the Houthis,” he said.
A major element of contention in the conflict has been the restrictions on the import of fuel and humanitarian goods into Yemen, where millions of people are on the brink of famine.
For years, the internationally recognised government accused the Houthis of controlling the distribution of supplies from the port, allowing them to decide who will benefit from foreign aid.
The announcement comes as the UN envoy to Yemen, Hans Grundberg, has called for a truce over Ramadan, which is likely to start this weekend.
Mr Grundberg said he was engaging with warring parties about the ceasefire.
The Saudi-led coalition announced on Tuesday it would begin a ceasefire at 6am on Wednesday to enable the political talks to take place.
The Riyadh summit is expected to continue until April 7.