Must a Priest Process to Altar at Beginning of Mass?

Yes, as the General Instruction of the Roman Missal (GIRM) provides, priests are to process to the altar at the beginning of Mass:

Among gestures included are also actions and processions: of the priest going with the deacon and ministers to the altar; of the deacon carrying the Evangeliary or Book of the Gospels to the ambo before the proclamation of the Gospel; of the faithful presenting the gifts and coming forward to receive Communion. It is appropriate that actions and processions of this sort be carried out with decorum while the chants proper to them occur, in keeping with the norms prescribed for each (GIRM 44, emphasis added).

A. Mass Without a Deacon

The Introductory Rites

Once the people have gathered, the priest and ministers, clad in the sacred vestments, go in procession to the altar in this order:

a. The thurifer carrying a thurible with burning incense, if incense is used;

b. The ministers who carry lighted candles, and between them an acolyte or other minister with the cross;

c. The acolytes and the other ministers;

d. A lector, who may carry the Book of the Gospels (though not the Lectionary), which should be slightly elevated;

e. The priest who is to celebrate the Mass.

If incense is used, before the procession begins, the priest puts some in the thurible and blesses it with the Sign of the Cross without saying anything (GIRM 120, emphasis added).