All the Starship air-starts up to this point have been landing burns using small header tanks which gave less ullage requirements, but I can't recall any mention of how Starship settle it's propellant in the 8 second coast after booster shutdown. Whether the engines will be started using the header tanks, or using ullage rockets.
I think we know the answer to this question now. According to Musk's June 25 Twitter Spaces chat... SpaceX plan to hot stage for the next orbital launch attempt.
So the upper stage engines will start before all the booster engines are cut off. This gives continuity of momentum and therefore upper stage ullage will stay at the top of tanks and the propellant will cover outlet to the engines, a technique the Russians were quite fond of.
Before launch the Starship tanks are pressurized to 6 bar via the Ground Support Equipment. After launch ullage pressure is maintained by autogenous pressurization. A proportion of the hot oxygen and hot methane gases are directed from the Raptor engines directly into their respective tanks to maintain pressure as the head space in each tank expands.
This hot gas is not very stable in contact with the cyrogenic liquid propellants but under acceleration the gas stratifies and the liquid is forced to the bottom of the tank. But when the engines shut down at stage separation the cryogenic liquid is free to float around inside the tank and the high temperature gas is rapidly cooled and depressurized.
To start the engines again it is necessary to re-pressurize the tanks from compressed gases stored in large Carbon Overwrapped Pressure Vessels (COPV). The problem is that when the tanks are mostly empty this takes a lot of gas and especially so because the COPOV supplied gas is cold and so much more dense than the hot autogenous gas (around 14 tonnes required).
With hot staging the Superheavy booster does not shut down its engines. They are merely throttled down and Starship ignites its engines to pull away so both stages remain under acceleration continuously during stage separation. After separation Superheavy conducts a flip maneuver and boosts back towards Boca Chica.
Unlike Falcon, Superheavy is made of stainless steel and does not require a re-entry burn and the landing burn is powered from the header tanks so there is never a need to re-pressurize the main tanks after boost back saving a lot of mass in COPV stored gases.