Mostly you don't roleplay it, you ask what they want to do in a set time (a week, a month) then judge what are the results or ask for specific rolls. Unless you want to make downtime a primary gametable activity.
You can start by asking how much time they will spend in downtime, or you can give a set time if a future event its known...
Ex 1:
GM "you've reached the safety of the town, what do you want to do and for how long?"
Magic User "I want to learn a couple spells, It should take me 3 weeks"
Cleric "I have some matters at my temple, I'll be off for 2 weeks, then I'll wait for you (the MU) to finish your doings" (I will tell you later -the GM- what I'll do in my last week)
Fighter: I'm going to see the boys at the garrison, then going to see the weaponsmith, I'll see if he can forged me a new weapon with this rare metal we found, it should take no more than a week, I'll be doing some training the next 2 weeks"
Thief: "Ok I'm going to talk around looking for gossips and see what happens around, maybe I'll do some quick and easy jobs
"
Ex 2:
GM "you've reach the inn you are staying, you know you have 3 weeks before the great meeting about the northern swamp exploration, what will each of you do in that time?"
Once you know what they want to do (players don't need to tell you in advanced everything they'll do) you tell them what happens, if a roll is required or if some roleplaying must be involved.