Start with a basin, open to the air. Run a pipe from a hole in the bottom of that basin to an airtight air supply container. (Glass flasks are shown here, and serve the purpose well. Cans with airtight lids may also be used.) The air supply container stands significantly below the basin.
Run another pipe from the top of that container up to the top of the airtight fountain supply container, which is filled with water.
From the fountain supply container, have a pipe that reaches nearly to the bottom of the container, through the top up to the spout of the fountain. A refinement is to have the spout coming up through the centre of the basin, and the containers may be concealed in a column supporting the basin, as long as the air supply container is lower than the fountain supply container.
Fill the basin with water.
The water from the basin flows by gravity into the air supply container. It fills the container, displacing the air.
The air flows into the fountain supply container,......