Welcome!
First of all you need to check the coolant water level is ok, preferably when the engine has really cooled down, and top it up as required. I know on my mum's Jazz (2004) the fan also runs a lot even though coolant level is ok.
Secondly have a look around the engine bay and see if you have any signs of water leaks. If the water pump is leaking you often see the white dried residue on the underside of the bonnet where it is sprayed up. With the engine running have a look to see if you can see any water spraying/running out anywhere.
Check the hoses to see if any look split or degraded.
I wouldn't think air is getting in to the coolant system, possibly just the previous owner never topped it up, and the low level of coolant starts boiling.
Just keep an eye on the level over the next days because normally you shouldn't need to top it up again if the system is watertight.