1. Question Restatement:
According to Jewish Tradition, or Mosaic Law, what was happening at the 9th hour?
2. Context - The 9th Hour of the Day, is 3PM :
... In both Jewish and Roman time keeping systems.
See : Re. The Crucifixion: Possible to Correlate Timekeeping and Calendar Days?
3. Answer 1 - They May have Chosen the 9th Hour, Because Nothing was Going on :
It is very likely that Peter and John - well known as Christians - were going to the Temple at the 9th hour - because nothing was going on, so as to not cause contention.
According to Acts 2, Shavuot - Pentecost - had already occurred, (Wikipedia; Exodus 23:16, Exodus 34:22, Numbers 28:26, Deuteronomy 16:10, etc.).
No other holidays seem likely, in this context.
Neither Morning, (Shakharit), or Evening, (Maariv), prayers would have been going on.
However - The Afternoon Service, (Minkhah), was arguably observed at any time - after 12:30pm, up to evening, as people were able to come, (because of work, etc).
4. Answer 2 - There is no Way to Know :
There are too many forgotten and conflicting traditions to know for sure : the Palestinian and Babylonian Talmuds (dated much later); the conflict between the Sadducee and Pharisees; the Karaites; etc;
However, I would venture a guess that the Sadducees were still in authority, in the Temple. But, they - and their traditions - have been purged from history.
Jewish liturgy, (see Siddur), throughout history, has been "organic". There are many of these liturgies today, all of which were written - because there is no Temple. None of them claim to reflect that period, let alone claim to have been passed down from Moses.
There is no way of knowing exactly what was going on at the 9th hour on a daily basis, though if some "event" was occurring, it likely would have involved sacrifices. Perhaps start with: Tractate Zevachim, “Animal Sacrifices”, Bava Metzia 109b, Shechitat Kodashim, “The Slaughter of Consecrated Animals”.