Yes, a Melkite Priest can celebrate Mass (which would be called Divine Liturgy) - in fact, a Latin Rite (or any other) Catholic can attend any Liturgy in any other Catholic Rite and fulfill his/her Sunday obligation. Likewise, that Melkite or other Catholic priest can celebrate Sacraments for you regardless of whether you are a Latin, Melkite, Maronite, Byzantine, or any other type of Catholic. A Latin Rite Catholic may attend a Melkite Liturgy and receive communion, and may have his/her confession heard by the Melkite priest. A few priests, including Fr Mitch Pacwa, have faculties to celebrate Mass according to 2 different Rites.
As for sacramentals, I suspect they could bless them, too, as I believe they can bless anything at their discretion - some of the Latin sacramentals just are not rooted in Melkite tradition.