There are several Zero-G airplanes to experiment briefly microgravity without going into orbit. One is the A310 from Novespace (F-WNOV):
Novespace A310 F-WNOV (source: Jonas Wirz at Airplane-Pictures.net)
When a Zero-G flight is executed, fuel should take the form of a sphere in the tanks, like demonstrated in the cabin for a beverage.
Advertisement in microgravity (source)
This effect won't appear in negative G, subject to the design, fuel lines may still be immersed in tanks, this cannot be the case in micro-gravity.
The 0G segment is likely flown with power. First this is not a continuously descending flight as seen below. Second, friction will slow down the free fall anyway and must be compensated by some acceleration from the engines.
0G segment is not limited to the descent (based on source)
How does Airbus ensure the fuel is still delivered to the engines at the required pressure, without air?
Basically it says no modification is required because the lines are pressurized, and this is supported by a 1966 Nasa study (Use of aircraft for zero-gravity environment) which is not more detailed. The line being pressurized downstream of the pumps seems only one element of an adequate design. Upstream fuel will experiment the effect of micro-gravity and lines are no more immersed in the fuel. Air can enter and create a problem when reaching a pump or an engine. Pumps could be drained. Maybe the problem won't appear during the 0G segment, but it should appear later.
I'd like to have the details about how pumps, pressure, flow, or anything else contribute to maintain a good fuel supply to the engines.
This is the fuel system for the A310: