He did wake up in the future.  But between the present and the future, he's asleep in a spaceship.  You're going to die someday.  You're not dead now just because sometime in the future you will be.  In Fitz's future, he'll travel back to the present and die.  But in his present, he's asleep in a ship.
Think of it like this.  Say Fitz has a watch and Simmons has a watch.  They both start at midnight on Jan 1.  After an hour, Simmons is transported five hours into the future.  In the future, her watch still says 1am.  Fitz goes to sleep and sets his alarm to wake him up in 5 hours.  He wakes up and Simmons has just arrived.  Her watch says 1am, his says 6am (1 + 5).  They have some adventures for 2 hours.  Then they travel back in time to 2am.  Fitz's watch now says it's 8am (1 + 5 + 2).  Simmons' watch says it's 3am (1 + 2).  Fitz dies.  This does not erase all the time Fitz was asleep for 5 hours.  So Simmons goes to find him in bed, say it takes her an hour.  Fitz's watch now says it's 3am, which is the actual time.  Simmons' watch says it's 4am (1 + 2 + 1).  Fitz is at an earlier part of his life (3am) than when he wakes up in the future (6am).
The paradox I mentioned is that this means that Fitz cannot be there to wake up in the future at 6am.  In the show, it's similar because it would mean that Fitz couldn't be there to wake up in the future to meet the rest of the team.  However, there's a bigger paradox in that the Earth was supposed to be destroyed while Fitz slept, and it hasn't been.  Back to the Future explained this as a splitting of the timelines.  There's now one future where the Earth is destroyed and Fitz wakes up to join the rest of the team.  And there's a second one where the Earth isn't destroyed and Fitz can be found and woken before getting to the future.  The Fitz who died is from the timeline where the Earth was destroyed, and the other Fitz, the one still asleep, is in the timeline where the Earth was not destroyed.