Bismihi ta'ala

According to our traditions, there are different views as to how long Prophet Musa (a.s.) lived for. The ages mentioned have been 120, 126, 240, 250 years. Regarding where he is buried, there is no precise information that we have. One view says he is buried close to Bayt al-Maqdas. One view also mentions that his burial site is on Mount Nebo, in Jordan. 

As for how he departed this world, it was generally of natural causes, but the story of how he passed is also interesting. 

It is said that the Angel of death, 'Isra'il had a conversation with Prophet Moses (a.s.), asking him permission to take his soul, and the Prophet entered into a dialogue with the Angel, and his soul was not taken. 

A while later, Prophet Moses (a.s.) was walking in the desert, and he saw a person digging a grave, and offered to help. After helping, the person asked Prophet Moses (a.s.) to lie in the grave to see if it fits, and he did so. When Prophet Moses (a.s.) lied in the grave, he was granted a vision of seeing his place in heaven, and then realised this person was 'Isra'il, and it was now time for him to depart this world, in the very grave that he dug for himself. 

This is how the story goes, as one narration says, and for the details please refer to this site:

https://www.imamreza.net/old/eng/imamreza.php?id=10038

With prayers for your success.