The orations for the priest himself (Orationes diversae, 8, 1962 [In previous editions of the Missal it is orationes diversae, 20]) are what one uses to say mass for a priest anniversary in the extraordinary form of the Latin rite. There is no special mass for the occasion.
One can say the mass of the day or pick a votive mass and add the orations under a single ending. The days that it is not permitted are specified below. Today being my anniversary, I'll probably say the mass of the day, which is a rogation day (the second rogation day before Ascension)!
Below is what the 1962 Missal says in that section (towards the back of the Missal).
ORATIONES DIVERSAE
8 PRO SEIPSO SACERDOTE
In anniversario propriæ Ordinationis sacerdotalis, quilibet sacerdos orationi Missæ, sub unica conclusione, orationem pro seipso addere potest, modo ne occurrant festa Nativitatis et Epiphaniæ Domini, Triduum sacrum, dominica Resurrectionis, festum Ascensionis Domini, dominica Pentecostes, festa Ssmæ Trinitatis, Corporis Christi, Cordis Iesu et Christi Regis, necnon Commemoratio omnium Fidelium defunctorum.
Quoties impeditur, oratio pro seipso sacerdote transferri potest in proximiorem diem similiter non impeditum.
In anniversario propriæ Ordinationis sacerdotalis, quilibet sacerdos orationi Missæ, sub unica conclusione, orationem pro seipso addere potest, modo ne occurrant festa Nativitatis et Epiphaniæ Domini, Triduum sacrum, dominica Resurrectionis, festum Ascensionis Domini, dominica Pentecostes, festa Ssmæ Trinitatis, Corporis Christi, Cordis Iesu et Christi Regis, necnon Commemoratio omnium Fidelium defunctorum.
Quoties impeditur, oratio pro seipso sacerdote transferri potest in proximiorem diem similiter non impeditum.
N.B. The 1962 Missal also has a note in the general rubrics, nos. 451-452. Previous editions of the Missal have that respective note in the Additiones et Variationes in Rubricis Missalis, VI, 8), 3.
P.S. Under this section of orationes for masses there are orations for such things as pro familia, ad postulandam humilitatem, pro devotis amicis, pro inimicis, etc. And under the section following the Masses of the dead there is a similar section of orations for masses of the dead of specific persons, e.g. pro defuncto sacerdote, pro patre et matre.