Can we predict when Solar System objects will occult a star?
Yes, we can predict when occultations occur.
The positions of stars are contained in a number of star catalogues. It is rather complicated as stars can have a number of different names, depending on which catalogue is being used. For example Rigel in Orion is also known as Algebar, Elgebar, β Ori, 19 Ori, HD 34085, HR 1713, HIP 24436 and SAO 13190.
It is possible to search star catalogues for all stars in a particular area of the sky.
The positions of planets and other astronomical objects can be obtained from ephemerides. The positions can be calculated for any given time.
An occultation can be predicted but it requires a detailed search. If the path of a body is calculated over a period of time, then a search of star catalogues can identify any stars which fall in the path.
The Moon is the best choice as it is relatively large and fast moving. It occults stars quite frequently and its position can be calculated from ephemerides.