I'm writing a bit of back story. Before my character became a shadowrunner he was a plain SINless street bum. Assuming he lived on the streets but had no connection to Shadowrunners or Fixers or Johnsons, is it still reasonable to assume he would know where to go to get a piece of cyber installed off the books?

Depending on the sprawl and city he was a bum in. But usually homeless people (irl and sr) are extremely resourceful (or else they would be dead), and it is easy to assume they would know where medical facilities are (for sellin blood/off the book testing etc) as well as where they could bring te body of Joe from the alcove next door after he bit it for some extra nuyen. Plus a lot of chopshops would advertise I would think, how else do people know they are there, and even if the advertising is word of mouth, the homeless are the eyes and the ears of the streets.

My question would e how he got the cyber? (Paid for by someone else, reward for doing unsavory work, Joe had a cyber arm and a layer of dermal armour and in exhange for the body and arm the bum gets the dermal, etc)

for a lot of simple basic implants, there will be charities that help the needy. so you could explain a cybereye or limb as medical care from an organization dedicated to helping the homeless. some doctors do free clinic work after all, and its very setting-appropriate to have such people trying to help. and if the limb is 'used', then you could use that as an excuse to add in fancier stuff that may not be entirely necessary.

so, for backstory i would have him get the bloody crap beaten out of him by someone- possibly gangers, possibly corp security depending on how you want to tell the story- and he winds up getting help from a street doc who does some charity work. maybe he got hit by a car. heck, maybe he got in the crossfire of a shadowrun, and one of the runners got him to help because they thought it was the right thing to do. slap on some secondhand alpha cyberware and your good to go.

how much cyber are you thinking of overall?

For more exotic/quirky cyberware and a bit of Southern flavour, he could be from Atlanta, specifically the Southtown area around the old Fulton County Stadium (Neo Anarchist Guide), where since the mid 40's various corps have been using the domer homeless in the area for field testing experimental cyberware and bioware, so could be a good way to have picked up something unusual....

Also, don't forget that there are plenty of street-level cyber clinics out there in SR. You can go to the local mall for a nose job, an ear implant, or a cybereye over your lunch break.