Someone in my office has written this email.
I am on leave after 2:30pm today. Have a doctor's appointment for my son.
Is the above sentence correct?
My correction would be -
I will be on a leave after 2:30 pm today. I have an appointment with doctor for my son.
The first sentence is using the present tense to describe a current state of affairs, even though the action is in the future. We can use the present tense when future events are seen as timetabled, or inevitable. Examples
It would be equally correct to use a future tense:
The second sentence has had the word "I" cut from it. This is an example of ellipsis of the subject. It is more common in spoken English, and is normally avoided in formal written English. The use of the possessive (doctor's) is natural and correct.
Your second sentence is not idiomatic, since the word "doctor" needs an article. You could say "I have an appointment with the doctor...", however the original version means the same and is shorter.
In the first sentence
I am on leave after 2:30pm today.
your office coworker is saying they will be out for an extended period of time ("on leave"), beginning at 2:30pm in the afternoon.
In the second sentence, your coworker is telling you why they are out
Have a doctor's appointment for my son.
They need to take their son to the doctor's.
The listener could easily assume that both are related, and possibly your coworker's son has a serious condition since your coworker is taking "leave" (possibly "medical leave") to look after their son.
If your coworker was only leaving for the afternoon because of the doctor's appointment, they might usually say
I am out of the office today after 2:30pm.
I am leaving the office at 2:30pm.
I will be out after 2:30pm today.
It might be that their son is fairly young since
(I) Have a doctor's appointment for my son.
implies the appointment was made by the parent, instead of
My son has a doctor appointment.
