my teacher advised no rubato, but rather a more 'flex' in the playing that is sort of unperceptible. of course, when i played it at the beginning, it still sounded sort of midi as you said. but, later on, it was like a kite blowing in the wind. you can use dynamics to your advantage to make the kite feel like it is gently blowing up and down.
also, (not that i am as expressive as my teacher) there is a natural feel that my teacher has for certain places that harmonically take a certain turn. a very very slight difference of approach (or slight slight delay) without losing the general beat is helpful. my teacher looks for 'tritones.' this was a real revelation to me, because i always looked for one harmony to move to the other.
also, when you do the repetitive measures at the bottom of the second page, you can focus on the alto voice the second time, if you want (for a little difference). i believe the more you play a piece, the more it 'becomes you.' your emotion may be different than someone else's emotion (we don't all have the same levels or the same reaction to a piece). don't feel badly if you think you don't 'sound as good' as someone else. just make it your own real feelings.
for me, i would think of my husband (or you can think of someone that YOU love and care for). the passages at the top of page two are quaint and intimate. to me, almost like brahms was saying if i lose you (your breath), i've lost my own (breath). i'll follow you anywhere. i play the question/answer phrase with only the slightest break between phrases (like a violin - as the bow has to retouch - but it doesn't take that much time). if you think like you are a violin, you'll probably have an idea how the phrasing should sound. you could even get someone to play it on the violin or have someone sing it with nonsense words - or sing it yourself (la la). it sounds silly, but quite realistic according to the favor brahms showed the violin and voice. set your metronome first and try to keep it even, then for a month or so after you've learned it, just do minor adjustments to the 'flow' that make it sound musical without losing the 'line.'