I can't find much information about crossing by foot and opening hours. I know it's allowed in the day, but I don't know whether they close that option at night. There is a second land crossing between Singapore and Malaysia, namely the Malaysia–Singapore Second Link (Google Maps), but it's not allowed to walk across it.
All three land border checkpoints operate 24 hours a day. As far as I can remember, ICA, on the Singapore side, "doesn't encourage" travellers to use the Causeway, but the access – stairs and a one-way turnstile – is opened 24/7. It is indeed faster sometimes, when traffic is bad. But at night, I don't see the attraction in doing so.
Malaysia Ministry of International Trade and Industry, Frequently Asked Questions (FAQ) Reopening of Land Borders (Vaccinated Travel) between Malaysia and Singapore (2022-03-25):
Pedestrians are permitted to walk across the Causeway and Tuas Second Link.
I can find no specific prohibition on walking at night so I assume walking is perfectly legal at all hours including at night.
It is common during peak hours (when walking can be faster than any vehicle):
For long stretches, there is no pavement/sidewalk so you have to walk on the road, which can be dangerous especially at night.
Comment on Second Link: The Wikipedia page claims that "travellers are not allowed to walk along the lengthy Second Link" but this is false or at least contradicts the above cited FAQ.