Ok, +5 for basing this question off Mass Effect (that’s just awesome).
I think it’d depend on how much the information affected me personally, and whether or not I could do something about it. If it’s going to affect me greatly and I can’t do a damn thing about it, then at least let me get through the day at work before nailing me with the information… and yes, have a heart and do it in person (unless you live thousands of miles away, in which case, at least pick up a phone instead of texting). But, if it’s a sensitive issue of some urgency which I can do something about, absolutely: let me know asap by any means necessary.
That having been said, I had a casual acquaintance years ago who called and left a message on my answering machine (you know, those things that existed before cell phones) to tell me that my (now ex) boyfriend had cheated on me the night before. She literally giggled about it. Uh… appreciate the info, but on the answering machine? Really?