This blessing is not indulgenced - at least, not “explicitly”. There are several forms of the “apostolic blessing”.
Any blessing that the Pope gives is called an “apostolic blessing”. This acknowledges the fact that as Pope, he has succeeded to the office of St. Peter and the exercise of that apostolic office.
However, when people say the “apostolic blessing” they usually mean one of two things:
(1) The Solemn “Apostolic Blessing”, that is the papal blessing with the plenary indulgence that is given by the Pope “Urbi et Orbi” (to the city [Rome] and the world) on the great feasts of Christmas and Easter and on extremely solemn occasions. This blessing may also be given on a solemn feast, three times in the year by a bishop (or, in the case of a special indult, a priest, according to the terms of the indult).
Wikipedia has the text for the Urbi et Orbi blessing.
When the bishop gives the blessing there are two rites to be observed:
After the Pentiential Rite, the “May almighty God have mercy” is replaced with a special formulary (similar ot that of the Urbi et Orbi blessing)
The text of the Final Blessing is:
Through the intercession of the blessed apostles Peter and Paul, may almighty God bless
you, the Father, and the Son, and the Holy Spirit.
(2) The Apostolic Blessing in danger of death, which may be given to the dying by a priest in the name of the Pope. The formularies (there are two in the post 1970 liturgical books) are:
Through the holy mysteries of our redemption,
may almighty God release you
from all punishments in this life
and in the life to come.
May he open to you the gates of paradise
and welcome you to everlasting joy…
By the authority which the Apostolic See has given me,
I grant you a full pardon
and the remission of all your sins
in the name of the Father, and of the Son, +
and of the Holy Spirit.
Both these blessings, (1) and (2), are indulgenced with a plenary indulgence.
What about the “apostolic blessing” of the letter, or during addresses of the Pope, etc.? These are simple blessings, like any priest may give: they are called apostolic for the reason that I mentioned above. You can think of them as “private” or “non solemn apostolic blessings”.
The indulgence attached to these blessings is only a general indulgence that applies to the devout reception of *any *sacramental of the church. That is, unless it is specifically mentioned or decided that a particular blessing will be indulgenced in a special way.
The Pope does not need to pronounce the words and make the sign of the cross. He has the intention and the prayer for the people to whom the blessing is addressed. Normally this is communicated visibly through the gesture. In this case, the visible communication is in the letter.