Dimidium facti qui coepit habet.
(He who makes a start has half the work done.)
                                                 Horace

The best way to start a letter is to show the date and then write "Dear Mr. Shakespeare..." Since Shakespeare claims in many of his sonnets to be immortal, you don't have to worry about the fact that many centuries have passed since he wrote Sonnet 73.

If I were writing such a letter, I would say something like this.

Dear Mr. Shakespeare,

I just wanted to write and tell you how much I like your Sonnet 73. I have read it many times and always find something new to admire. I especially like the way way you use simple imagery drawn from nature, such as:

"bare ruin'd choirs where late the sweet birds sang"

I have observed that the images may seem simple but that they are really very intricate. In each stanza you have a metaphor and then a metaphor for the metaphor. 

This is my suggestion about how to get started. You can just express your appreciation of the sonnet and pretend that you are addressing William Shakespeare. You might close your letter with something like:

I hope this letter finds you in good health at Stratford on Avon.

Respectfully yours,

I don't know how long your letter has to be. You can find plenty of information about Sonnet 73 in the eNotes Study Guide because it is one of Shakespeare's most famous sonnets. (See reference links below.) What makes Sonnet 73 so impressive is the dazzling display of imagery and also the characteristic simplicity of Shakespeare's images.

You could mention other works you admire, such as Hamlet and Macbeth, or other sonnets. As long as you have his attention, you might ask him a few questions about his works which have puzzled scholars for centuries. For example: Was Hamlet really mad? Ask him anything you like. Maybe you'll even get a reply!