Now, I know that at least several players tried to complete the game as fast as possible - that is, with minimal 'Days Passed in Game' value (along with a minimal 'Playing Time' value).
Of course, there is no ways to reduce the 'Playing Time', since it reflects the total amount of time you spent playing the game (NOTE: in real-time equivalent! [i.e. 24 in-game hours = 24 minutes of Playing Time]), but there are some ways to make your 'Days Passed In Game' stat lower than it could be.
Here are some advices for that:
1. For experienced players only: Save as rarely as you can. Do large chunks of storyline missions and/or side missions in one go, without saving. Of course, this increases the risk of you getting wasted or busted with every single additional minute you play, but if you can take the risk to last long enough and get to a save point in the end, then you're really skilled at that game. Thing is, saving the game advances the in-game clock 6 hours. This means that 4 saves give you an additional in-game day passed already. The less you save, the less days you 'spend'.
2. Decrease your movements around the map: Moving around the map takes time (pretty evident, eh?) That is why, to reduce time spent on travelling across the map, I suggest you complete missions or other stuff like Hidden Packages/Rampages etc. in one patch of the map, then move over to another one and so on.
3. Find an appropriate means of transport: Vehicle choice is also crucial. Helicopters can take you anywhere you want fast, but you will lose some time starting their rotors or landing them appropriately. Bikes are fast (even the Faggio) and are easy to mount, yet you will be prone to falling off on even a slight collision - and that again adds some time to your playthrough - trying to mount back on the bike, only to fall off again. Cars are good, but they can't be used to make shortcuts through many narrow alleyways... So the ultimate decision is up to you. And of course, I hope you won't be going around the city in a Coach or a Flatbed - those things are SLOW and HUGE!!!
4. Utilize the in-game time loops: There are some missions in which the time is preset automatically, and the clock will adjust to that specified time each time you start that mission. The only two I can think of now is "Bloodring" and "RC Bandit Race". Whenever you start those missions, the in-game time back will loop back or forth to 10:00 of the current day. Obviously, if you accept the "RC Bandit Race" mission at, say, 01:00 (luckily, not possible for "Bloodring", as it is available from 20:00 to 23:59 only), you will lose 9 in-game minutes and will advance the coming of a new in-game day... My advice would be to accept those missions at around 23:50 (not later, since the mission loading process may take some time and you'll lose in the end - if mission is initiated when the clock has hit 00:00). That way you can save up to 13 minutes 50 seconds of in-game time (meaning your in-game day will last 37m50s instead of regular 24m0s).
That is all advices I can think of right now. If I discover something later, I will be sure to add it here (that is, if the topic deserves to live long enough 'till my return)