I'm not sure how anyone could interpret Friar John's actions as apathetic. He could not deliver the letter, for the reasons that Post 5 makes clear. It could be argued that if he had been aware of the importance of delivering the letter, he might have risked more to get it to Romeo. But I think it's pretty harsh to say he didn't care. In any case, as others have pointed out, his character is not really developed at all, as opposed to Friar Lawrence.
We know very little of Friar John's personality. Certainly he is willing to help Friar Lawrence so we might say that he is obedient or kind. He is caught up in a quarantine so we know that his inability to deliver the letter wasn't because of cowardice or bad intent. All we can say for sure is that Friar John tries to help Friar Lawrence. He has no idea what is in that letter. We know that he does make the attempt but that is about all we know. It is difficult to draw conclusions about Friar John's personality from our limited contact with him in the story.
Friar John, not to be mistaken for Friar Laurence, is the messenger that Friar Laurence sends to Mantua. However, when he arrives with the message for Romeo that Juliet is not dead, but simply in a type of trance, the little monk is turned away because the town is under quarantine. Obviously, he seems very meek as he makes no further effort to communicate with Romeo.
Friar John definitely cares about Romeo and looks out for him. He seems to be a mentor and a kind of father figure. He still agrees to marry Romeo and Juliet, but he has an ulterior motive. He wants to end the feud. In the end he feels grieved for the part he feels he played in their deaths.
Friar John is not a well-developed character in the play, though his role in developing the plot is important. He is responsible for carrying Friar Laurence's letter to Romeo to Mantua, but he is unable to do so due to fears of the plague. He goes to get the crowbar for Friar Laurence, who plans to use it to open the Capulet crypt. These are his only actions in the play. We do not discover anything about his personality, except perhaps that he is loyal and obedient to Friar Laurence.
I feel that Friar John has an attitude that portrays a "I don't really care" kind of feeling.