how to underestand "of" in "starting times of half past eight for high-schoolers"?
No; it's the usage shown here:
of preposition (THAT IS/ARE)
the problem of homelessness
a rise of two percent in inflation
the skill of negotiating
the difficulty of bringing up twins
the pain of separation
At the age of six she could read a newspaper.
[Cambridge Dictionary]
The second example given is perhaps the closest that queried. ' ... a limit ... of half past eight ...'.
CED seems to outperform OALD here in listing different senses.