How does Shakespeare assure his friend that his beauty will ever remain undying?

Shakespeare asserts that his friend will be immortalized through Shakespeare's verse. The words in his poems will insure that his friend's beauty will remain undying. As Shakespeare writes in Sonnet 55:

So till the judgment that your self arise, You live in this [the poem], and dwell in lovers' eyes.

Shakespeare also notes in the above quote that the friend will live in the memories [eyes] of his lovers.
Shakespeare repeats the idea of immortality through poetry in Sonnet 81:

Your monument shall be my gentle verse, Which eyes not yet created shall o'er-read.

As indicated in the first quote, this is not a pagan concept, as it might be in Beowulf, where the hero's only immortality comes through verse or art. Shakespeare is careful to note that the verses he writes simply act as a placeholder to keep the friend's memory alive until the Second Coming raises him again in bodily form, a Christian concept.
The concept of art as immortal in contrast to the decay and death of the human body is a common one in poetry. Often, however, as with the Roman poet Horace, writers thought of the lines they wrote as immortalizing themselves. Shakespeare, however, puts the emphasis on his poems keeping alive his friends and lovers. We may nevertheless suspect that he was also thinking of gaining his own immortality through his writing.

I assume you are referring to Shakespeare's "Fair Youth" sonnets (1–126) with this question. These sonnets were written to an unidentified young man beloved of the poet, and many of them focus upon the themes of youth, reputation, death, and beauty. The poet does assure his beloved on several occasions that, although "summer's lease hath all too short a date" (Sonnet 18), his beauty will not disappear with his youth. In Sonnet 18, the poet states, "thy eternal summer shall not fade," for the simple reason that "so long as men can breathe, or eyes can see, / So long lives this [poem], and this gives life to thee." That is, the poet is sure that his verses will live on forever, and therefore the youth and beauty of the young man are immortalized in them.
This theme is continued in Sonnet 54, for example, where the poet compares the beauty of his beloved to that of roses, which, in dying, simply produce beautiful odours to replace their beautiful flowers. Here, Shakespeare assures his "beauteous and lovely youth" that "when that shall fade, my verse distills your truth."
Shakespeare's sonnets repeatedly revisit this theme of verse as an epitaph. The poet is concerned that his works should allow the continuance of his own reputation, as well as of the beauty and reputation of his beloved.