You get this mission by going to the Malibu and talking to Kent Paul.
This is considered the hardest story mission in Vice City, as there is a time limit (Lance's health will drain away gradually and you fail if he expires), and its multistage, so preparations are crucial.
Before you accept this mission, stock up on full health, armour, and weapons, especially Uzi ammo. Memorize the quickest route to the junkyard. Also, get a fast car or motorcycle (I prefer the Freeway) and park it outside the Malibu, but not too close or it’ll disappear before you trigger the mission.
Get to the junkyard as quickly as possible, where the [Pink Dot] marker is. There is an initial roadblock at the junkyard gate, so stop in the alley and use some heavy weapon to blow up the car, which will take a few of Diaz's men.
Then drive to the hanger where Lance is held hostage, avoiding bad guys until you get inside, where you have no choice but to kill the 2 thugs guarding him. You might have to also take out some thugs stationed just outside the hanger that will follow you inside.
I hope your ride is a bike or you can also take a panzer ( Rhino ) as it is less of a target than a car, and the bike can go inside the Hanger, this way you and your vehicle aren't fired on by all the thugs outside when you disembark. The Freeway is a good choice, as it is fast, tough, and decent on rough terrain. The PCJ600 would have been tripped up by the junkyard grounds, and the Sanchez's off-road performance would not have compensated for its mediocre speed.
The enemies will drop lots of cash in this mission. However, as there are so many of them outside and as quite a few have good vantage points, (i.e. up on the crane) where they can hit you but you can't easily hit them. As well, Lance's fading health doesn't afford you much time to take them out. Though you might feel cheated at no reward for this mission, the next mission will make up for all of that.
Now to get Lance to the downtown hospital. Some contributors suggest using the Sentinel to the left of the hangar, which I would avoid unless your own ride is badly shot up, particularly if a tire is blown out or the engine is smoking.
Right as soon as you leave the junkyard gates, Diaz's men will go after you in three Comets, but you should be able to make it quite easily. In a bike, you can go in the narrow alleys of Little Haiti to lose the Comets, then it should be straightforward to get to the downtown hospital.
Alternatively, to get rid of the Comets easily, as soon as you leave the junk yard, make a right and go over the small bridge and wait there for the cars to come, you can then tempt them to go into the water, or just shoot them to blow them up easily.