A fountain pump is a small submersible motor with hoses and fittings that clog and leak if packed wet. Drain it, clean it, and bag the parts, and it drops right into the new fountain.
Pro tip: Clean the impeller intake now while you have it out; a clogged intake is the top reason a pump fails to prime later.
Watch out: Never handle or move a fountain pump while it is plugged in; unplug at the outlet first to avoid shock.