I think this definitely is not an intended answer, but I will try to suggest
What goes up, but not down.
Time always goes forward (i.e. time coordinate increases, or goes up, and never decreases (goes down)). Also, we say "time is up", and never "time is down".
Around, but not back.
Again, time never goes back.
Forward, but not fast.
It's arguable, but time never passes that fast.
Two steps to the right, and not usually to the left.
When plotting a time-dependent variable, a time axis usually goes from left to right.
You don't know me, but I know you.
a wordplay: watch (time-keeping device) / watching (surveillance)
I come out during the day, but I am harder to find at night.
Well, at night we must usually... sleep, so it's harder to find some time at night. Of course, this is arguable again.
I am not extinct, but I may be found in your sink (probably not though, but it rhymed).
When you are washing your hands, your watch can detach from your hand and fall into the sink (of course, it probably won't).