Which makes me wonder... if I want to pray the office say in the anniversary of burial of some Christian, do I have to pray vespers the previous evening?

The vespers of the prior day, or "anticipated" vespers? The quoted article speaks of vigils with regards to the feasts of saints, and also in the context of the practices of previous centuries. If praying the full Divinum Officium or Monastic Breviary, there's the actual vespers of the previous day, since AFAIK I don't think vespers traditionally are anticipated. Vespers are prayed at dusk, and the liturgical daily cycle traditionally begins a half hour after dusk; thus in practice it's actually the Matins of the following day that are sometimes prayed late the night before. Or did you mean the vespers of the prior day as in a vigil? (For this paragraph, Simeon is probably the better one to explain, or perhaps someone else might know.)

In contrast, the Little Office of the Blessed Virgin Mary is simpler insofar as the content generally does not vary much from day to day, except for the Nocturns of Matins and, more broadly, some of the Lessons and other prayers according to the liturgical season. For the associated Office of the Dead, the instructions are to pray Matins, Lauds, and Vespers after you pray those respective Little Office hours. It's common practice to pray Matins and Lauds together in the very early morning, so Little Office first on both, then Office of the Dead following on both.

Bottom line, no need to pray vespers the previous night if you're observing the anniversary of a burial. The rules might be different for vowed religious or third order oblates, but for laity in general, these devotions are of counsel rather than precept. No less, a good thing to do!

From the article:

And also:
I know that those quotes refer to the Papal liturgy and Dominican rite but they seem to indicate that vespers of the Dead are sung on Nov. 1st. Also, I think I read somewhere on that blog that John XXIII changed the All Souls' office so that Vespers were sung on Nov. 2nd.

Exactly. Modern clock-time begins the calendar day after midnight, i.e., 12:01 a.m. (Similarly, I'm up an hour early this morning due to that modern artifice of "turning back the clock".) In contrast, the traditional liturgical day begins at nightfall. Those late vespers sung on the calendar date of "November 1st" were after the second vespers of All Saints, hence actually belong liturgically to the observance of the following day. Note too that this first week of November is an octave, which is why indulgences for the poor souls are still possible for a few more days through 11/8. In effect, the observance of All Souls on 11/2 shifts the focus from Church Triumphant to Church Suffering for the remainder of the octave. As for that last mentioned blog, once we get to whatever John XXIII changed, then all bets are off, as it were.

Sorry if this is derailing the thread but I think it is relevant. I will pray the office of the dead again today (correctly Deo volente).

Not at all, this is helpful. Thanks for seeking out that clarification.