NRT is a nice airport to connect through but you'd be looking at a minimum of one more connection, most likely in BKK or SIN if you want to stay with *A (IAH-NRT-BKK/SIN-PER).

You could also try LH out of IAH, flying IAH-FRA-SIN-PER on LH and SQ but that would involve back-to-back redeyes and put you into PER sometime in the morning two days after leaving IAH. Or could perhaps fly IAH-LHR-AKL/SIN-PER on a mix of carriers.

There are lots of options but, if you want to stay on *A, few good ones.

DXB via EK sounds like a good choice. Sadly, as if UA wanted to give you a swift kick on the way out, they terminated their Mileage Plus partnership with EK last May.

Best of luck.