Yes, casting ritual transmutation spells will allow you to change the stone's effect
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.