Can you go under the hood & pinpoint the location of the noise? Sound like the ABS pump motor which is supposed to run & pressurize the ABS system at startup. It's normal for that thing to run for maybe 10 seconds, then it stops & the ABS light goes out.

You can bleed your ABS modulator to remove air, just having the reservoir full isn't enough. Still, being 11 years old, maybe a pressure switch is bad or ??

Do you know how to jumper the SCS connector for reading ECU error codes? Do the same thing, only it's the ABS lamp that you watch. Long flash & long pause are first, that's not part of the code. Count flashes for 1st digit, short pause, more flashes for 2nd digit of code. Long pause then flashes for the next code. I've got a code list for '95 Integra, I think it might be the same for your car but not sure.