Take the whole day. I’ve been on both sides—hiring manager and candidate—and the half-day medical appointment is a tell, especially in a small office. The more specific the excuse, the more it invites questions. A simple “I’m not feeling well and won’t be in” is a complete sentence.
Check your employee handbook before you do anything. If your company is covered by NYC paid safe and sick leave, you may have protected time; in practice, many small offices only ask for a doctor’s note if you’re out multiple days. One sick day usually doesn’t require documentation, but do not mention an interview to a coworker. I once saw a perfectly fine candidate sink himself because he told the receptionist he was “so busy at his current job” and she knew his current boss socially from a Brooklyn networking group. It’s a small town.
For the employment check: most background checks do not call your current employer until after an offer, and reputable firms ask permission before contacting current supervisors. When you fill out the form, mark “do not contact” for your current employer. That is normal and they see it every day.
Finally, do not update LinkedIn, do not post a photo from Midtown, and give yourself at least 45 minutes of walking time between the interview and the nearest subway. You’ll be more convincing if you’re not out of breath when your boss calls about a file.