How long was Vincent under Michael's tutelage? I think Mike worked with his father for a few years before he passed. Did Mike spend as much time with Vincent as he did with Vito to make sure everything was set when he was to make him the Boss?

Remember that GF3 takes place entirely in 1979 and 1980. So Vincent becomes boss within two years of the beginning of the movie. And up until he meets Michael at the party, it appears that Vincent hasn't had that much direct contact with him. Moreover, Michael clearly wasn't even conceiving of making Vincent boss at the start of the movie. He seems content to retain Joey Zasa as head of the branch of the Corleone family remaining in New York. In GFI, by contrast, Michael kills Sollozzo in 1945, comes back a few years later, and then is under Vito's tutelage for another few years until Vito dies in 1954. So Michael clearly had a much longer "apprenticeship" than Vincent.

Vito needed Michael to succeed him because Sonny had been killed, Fredo was out of it, and he was physically and mentally spent. Michael's apprenticeship began when he returned from Sicily (probably ca. 1947) and by 1954, when he visited Moe Green (car that takes him to the hotel is a '54 Imperial limo), the Don had "semiretired" and Michael was head of the family. But I think he downplayed that transition because, after Vito died in 1955, Michael wanted to appear weak--the new guy in charge--to trap Barzini. It was all carefully arranged.

By contrast, Michael was distainful of Vincent at the beginning of III, then rather cynically (IMO) began to use him as a kind of cat's paw for his foes. Only after his stroke did Michael seriously begin to regard Vincent as his successor. You could say that Michael's illness and Vincent's ascendency paralleled Vito's shooting and Michael's ascendency. But that'd be superficial, IMO. I think Michael's plan was for Vincent to take over the olive oil business and act as his pit bull and deterrent while he became "completely legitimate" through Immobiliare. And Vincent's quid pro quo was to give up Mary, whom Michael probably wanted to marry off to some wealthy, powerful and respectable European who could enhance his standing. He was using Vincent, all the way.

The time frame was MUCH shorter, and in fact that is a huge flaw in III.

The party celebrating Michael's getting into the Order of St. Sebastian or whatever took place in spring or summer of 1978.
The same night as that party, Vincent whacks Zasa's men and Michael tells him to work for him. within a matter of weeks Michael is bribing the church, getting their preliminary vote on the purchase of Immobiliare, and after the helicopter hit is in the hospital. There, Kay tells him that Anthony is already getting good notices (who gets good notices within weeks of announcing he is going to be in the opera?) AND that he will be starring in Caveliere Rusticana in Palermo on EASTER....which would be 8 or nine months away. By then
Michael is already set to hand the whle think over to Vincent who had been his apprentice for less than a year.

The party was actually in 1979--the caption at the beginning of the movie says "New York City, 1979."