I'm a writer working on a story set in the late 1800s where one character attends Mass on a regular basis. I've attended several of Masses myself (non-Catholic here but I come from a large Catholic family), but I'm not sure what to include from my own experiences as they may be anachronistic. Other than the liturgy being spoken in Latin, what else should I consider to achieve a level of verisimilitude in my work?
Formally, the kind of Mass celebrated before Vatican II (and still celebrated in many places) is called the Tridentine Mass. (You might find this conservative Catholic step-by-step guide useful in your work.
The main difference, as you've noted, is that it was in Latin.
Also significant is that the priest doesn't face the congregation from behind the altar. Instead, for most of the service, the priest stands with his back to the congregation, as if he were the leader of a group addressing the crucifix on the back wall (or the tabernacle underneath, in which the Eucharist is stored - which is to say, the physical stuff which will become the literal presence of the Almighty through the mystery of the Mass).
There are bits where the priest would address the congregation - primarily, reading from the gospels & epistles and, possibly, a sermon on the meaning of the readings (the "Homily").
There will also be more portions of the liturgy sung or chanted. Some of this will be things that the congregation will know responses to that they'll have to sing, like a responsorial psalm (the "Alleluia" is one part that's still sung by the congregation in Vulgate Masses today). In some parishes, there'll be a choir to do the heavy lifting on the congregation's behalf.
The congregation will also know when to stand, kneel or sit during various portions of the service - I don't believe that's changed too much.
In a High Mass (super formal... which is kind of how a Tridentine Mass comes across today, although all Latin Masses would not necessarily have been considered "High Mass" before the 1950s), there's going to be incense, which will be blessed by the priest and then used in the blessing of the bread and wine (and other priests, if more than one are celebrating the Mass) prior to the most solemn portion of the Mass, the Transubstantiation (when the Eucharist becomes the body and blood of Christ).
Note for your character - there will be some Masses during the week that will be a lot less formal than the Sunday noon Mass. If your character is going to Mass more than once a week, the differences between the celebration in mid-week and the celebration on Sunday will be a way of keeping track of time.
Should also be noted that Vatican II really just codified some changes that were already underway - the key document addressing the changes is Sacrosanctum Concilium.