By valid ticket do you mean you have a ticket number? Or just a booking? If latter, I don't think there is much you can do unfortunately. If former, you should be able to get SQ via SIN (and NRT if you want First) without paying extra. If you buy a ticket on UA with $, SQ will not reimburse you. If getting there as fast as possible is the priority maybe you can get seats in J on NZ?