Having recently just done this I wanted to write a quick guide for how long it will take. The servers for the MGS5 will Shut down 31st May 2022. It took me 21 hour of gameplay to unlock this achievement and this is coming from someone who has played this game through on another platform. You don't gain access to FOB missions till you have completed mission 22. Once that is done you can select FOB missions from the idroid when in the helicoptor. I would just select a event fob to avoid any reprisals from players.
My Advice to do this as Quick as possible would be this.
1) Ignore most of the side missions bar the interpreter ones it goes without saying you still need to do the ones marked in Yellow
2) Upgrade your idroid and develop C4 and a sneaking suit as soon as you can.
3) Upgrade your fulton device as soon as possiable make sure on the mission "An Angel With Broken Wings" you grab the transport specialist. He is one of the hostages in the ruins of the building on the second floor at the first site you get too. The building with the tank and jeep in front of it shoot the jeep drivers then move into the build quickly as the search teams will kill him.
4) Make D-dog and Quiet available for deployment as soon as you can. I found D-dog at the start of mission 4 near the landing site or the mission start beacon. Don't kill quiet when given the choice.
5) Collect all misson rewards from the menu as soon as you can (This function will be removed by the shutdown)
6) My Perfered loadout is a silent pistol non-lethal. A silenced automatic weapon, A rocket launcher, make sure to equip grenades and C4 as well.
7) Grab resource at every chance you get. Small packs are worth 100, White containers 750, Red containers 7,500 you need a upgraded fulton for the containers hence the need fot the transport specialist.
I would highly recommend going for the nuke achievements as well before the shutdown as building them requires a lot of resource access to which will be heavy cut.