Can I use this copper wire to add a ground to an ungrounded outlet?
Go for it

For a retrofit equipment grounding conductor, your case falls under NEC 250.130(C) point 2:


  (C) Nongrounding Receptacle Replacement or Branch
  Circuit Extensions. The equipment grounding conductor
  of a grounding-type receptacle or a branch-circuit extension
  shall be permitted to be connected to any of the following:
  
  (1) Any accessible point on the grounding electrode system as described in 250.50
  
  (2) Any accessible point on the grounding electrode conductor
  
  (3) The equipment grounding terminal bar within the enclosure where the branch circuit for the receptacle or
  branch circuit originates
  
  (4) An equipment grounding conductor that is part of another branch circuit that originates from the enclosure
  where the branch circuit for the receptacle or branch
  circuit originates
  
  (5) For grounded systems, the grounded service conductor
  within the service equipment enclosure
  
  (6) For ungrounded systems, the grounding terminal bar
  within the service equipment enclosure


So, your plan with the split bolt is good.  You can use green THHN without a conduit for this provided it's protected from mechanical damage and routed within the building structure, by the way, as per 250.120(C):


  (C) Equipment Grounding Conductors Smaller Than
  6 AWG. Where not routed with circuit conductors as permitted in 250.130(C) and 250.134(B) Exception No.2, equipment grounding conductors smaller than 6 AWG
  shall be protected from physical damage by an identified
  raceway or cable armor unless installed within hollow
  spaces of the framing members of buildings or structures
  and where not subject to physical damage.


If you do need to make an exposed run, though, a bare armored ground cable is going to be cheaper and easier to handle than having to run conduit just for a ground wire.