Shakespeare did not own a theatre building--not outright at any rate. He did contribute financially to a couple and got a share of the profits from them. Shakespeare never planned or built any theatre buildings or worked on their construction in any way.
If you are asking when Shakespeare started working in the theatre business, the answer is nobody knows because this is an event that occurred during what are called his "lost years" between 1585 and 1592.
Actors Working For William Shakespeare
He bought a part of the globe theatre
It depends what you mean by "Shakespeare's theatre". Do you mean the theatre which was built in 1996 and is called Shakespeare's Globe Theatre? Or do you mean the theatre company which he joined, the Lord Chamberlain's Men, which was founded in 1594? Or do you mean the tradition of Elizabethan theatre of which Shakespeare was a part, which started in about 1560? It could be any one.
the theatre William Shakespeare built. The theatre William Shakespeare built in 1599.
American Shakespeare Theatre was created in 1955.
Unfortunately the people who worked there are all dead and left no memoirs.
Shakespeare did not have a theatre in Stratford. There's one there now, The Royal Shakespeare Theatre where the Royal Shakespeare Company plays, but there wasn't one in Shakespeare's day.
Shakespeare bought share in a theatre group where worked for five years. The name of the theatre is Globe theatre.
It's a Theatre someone has decided to name after Shakespeare. The most famous of these, although neither is strictly speaking called "the Shakespeare Theatre", are Shakespeare's Globe Theatre, built in 1995 in Southwark, London, and the Shakespeare Memorial Theatre, originally built in 1932 and substantially renovated since, in Stratford.
If there was a theatre called "William Shakespeare Theatre", you will have to be a little more specific. Was there such a theatre built in Akron, Ohio in the 1930s? Or in Calcutta in the 1890s? If the theatre you are talking about is "Shakespeare's Globe Theatre", it is still standing, having been built in 1997. If the theatre you are talking about is the Blackfriars Theatre, in which Shakespeare acted and held a small share, it was demolished in 1655. If the theatre you are talking about is the First Globe Theatre, in which Shakespeare also acted and held a small share, it burned down on June 29, 1613. If the theatre you are talking about is the Second Globe Theatre, which was built to replace the first one in 1614, and which might have had nothing to do with Shakespeare, it was torn down in 1644.
It was the Theatre Shakespeare's company built in 1599 and where his plays were performed. It could hold up to 3000 spectators standing and sitting. It burned down in 1613, was rebuilt with a different roof and eventually was torn down in the 1640s.The Globe Theatre (called Shakespeare's Globe) isa replica of the 1599 theatre built in London in 1996 and used as a working theatre.
Your question is ambiguous. Do you mean when did Shakespeare start to write it, when did they start to rehearse it, when did a performance start? We can only answer the last one anyway. If the play was being played at the Globe theatre it started in the mid-afternoon.