I didn't find it strange. Grief is complicated. Sometimes the mind understands things the heart isn't willing to accept yet. He opens up to her and his mind aknowledges he's attracted to her, but when he goes to act on it, his heart kicks in - he still loved Hina; it feels like he's betraying her.
I can tell you, if its been a year, he's starting to hear it from friends & family - 'it's been a year; you should be over her by now.' and 'it's been a year; just forget her and find a new girl,' or even, 'it's been a year; get over it,' but it's not that easy. When you love someone deeply, you DON'T forget, you NEVER forget, you just start learning to live again without them. It's a hard step to take... letting go of the past and facing a future empty of their love and presence. Their presence is gone, but you can hang on to the love, and living people often don't understand this. Accepting that is a hard stage to get through because it's hard for the heart to say good bye forever, even if the mind knows it's wise.
I'm sure he told her it was none of her business because he wasn't ready to let go and didn't want to hear what he had probably already been hearing, yet again. It wasn't until he faced with the realization his grief was hurting someone else that he really looked at what he was doing. It probably helped that she told him he didn't have to forget Hina, and his mind knew it could love Nazu if his heart could just let go. It gave him the strength to try to learn to live again... to open his heart to the realization it could love again without betraying the love it had for Hina. She was able to get him to see beyond the grief stage he had stalled out in and get him moving/healing again, and it's something that really happens - that people will stall out and be stuck in some stage of grief for longer than expected - so I found his 'fickle' progression believable. Maybe a bit hurried, as you said, once Nazu started 'opening his eyes', but still, IMO, believable.
It was too jumpy or you can say far too rushed. It's like there are some missing parts. Why would he open up and then tell her it's none of her business? All messed up, I tell you. You could tell what they wanted to make by doing this manga BUT it did not mix up well.