airlines pay to join what is called a 'pooling agreement' if an airline operates to say LHR but its base is say JFK, it would be unfeasible to hold all spares at LHR. But if they are 'pooled' and so too are BA then you could get a part from BA without additional cost. Not every part is part of the pooling agreement though. Another way is to loan parts on an emergency basis, however this can be very expensive, you pay per day.
A starter failure whilst not unusual, doesn't happen that often these days, plenty of preventative maintenance to prevent this, so should it fail and you are not at home base and not pooled it can be difficult and expensive to procure parts.
how long it might take to fix etc.
Is it like a car starter, unbolt/replace etc (obviously it's clearly not).
1-2 hours to replace maybe depends on the engine.
bleed ducts need disconnecting, seals/drive shaft may need replacing, it is normally attached to the engine gearbox by what's called a quad ring which can be difficult. If the drive shaft has sheared it may be difficult to remove from the gearbox drive.