Is this sentence written correctly? If not how would you write it. Compro la camisa y una corbata.

Since your sentence doesn’t has a subject, I would specify who is doing the action
Example: Yo compro la camisa y una corbata.
Ella/el compro (he/she)
Ellos/ellas compraron (they he and she)
.
So it would be (YO) compro una camisa y una corbata
As you can see in the image you omitted the subject “I” (yo), it’s not necessarily but you could put it so the subject of the sentence is known, although it is understood in Spanish that the subject is I. (Yo)