Why is that a problem? The
SG Arrival Card is a
very short form. It doesn't ask tough-to-remember questions such as your P2 math teacher's name. And even if you don't remember your own NRIC number or date of birth, you have the option to use Singpass to retrieve those details. Or you can pull out your physical NRIC, or you can manually check the Singpass app for your NRIC number. Your date of birth is printed on the passport that you must carry to cross the border, and it's also printed on your NRIC.
That's OK! You're allowed to do that! Walk up to one of the kiosks and input your SGAC details there. (ICA knows that mobile networks are sometimes unreliable and that phones sometimes don't have any battery power left.) Or stand to the side of the escalator, pull up the Web-based SGAC, fill it out, submit it, then proceed to the gate. Or connect to the public Wi-Fi network that's likely available in the arrival hall. Or if all else fails stop by the staffed ICA desk, and they'll get you sorted.
Singtel had a significant service outage this past March. "Stuff happens." Fortunately ICA provides multiple viable options to help you clear immigration even if you don't have a phone.