The text for a School of Transmutation wizard's transmuter's stone is as follows (PHB p.119):
Each time you cast a transmutation spell of 1st level or higher, you can change the effect of your stone if the stone is on your person.
Does this mean that if a ritual spell such as skywrite (XGE p.165) was cast, the wizard would be able to change what effect their Stone has active?
The quote you've provided does not mention spending spell slots, it simply says that you must cast the spell. Using ritual casting to cast a spell is still casting a spell, so this would allow you to change the transmuter's stone's effect immediately after you finish casting the ritual spell (so long as the ritual spell is a transmutation spell).
Compare this to the divination school feature, Expert Divination (PHB, p. 116):
When you cast a divination spell of 2nd level or higher using a spell slot, you regain one expended spell slot.
Since this sets a precedent for specifying that a spell slot must be used, it's clear that the transmuter's stone does not require a spell slot to be used, only that a transmutation spell of 1st level or higher is cast, so ritual casting satisfies this.
Furthermore, this question about the abjuration school's Arcane Ward feature (linked by @Sdjz) and the answers to it show that Sage Advice claims that ritual spells do recharge the arcane ward, which uses similar wording to the transmuter's stone (meaning it doesn't specify using a spell slot like Expert Divination does). This further supports that RAW changing the effect of the transmuter's stone doesn't require expending a spell slot.
Yes. Casting a spell is casting a spell, whether it is from a magic item that says "the user can cast blah blah once a day...", using a ritual to cast, using a racial ability or plain old using a spell slot.
The key is that it works if is the wizard that is casting the transmutation spell, whatever the source of the ability to cast.
