It seems your friend doesn't have a very good command of English. You asked the correct question to begin with, and he should have responded with not only the date but also the time of day.I would like to ask about the exact time of doing something, so I asked "what time will we do...?" in email, However, my friend answered "on 14th May", but I need the exact time like on 14th May at 5:00pm, should I asked "what's the exact time of doing..."? or whether there are other saying?
Since your friend doesn't understand the phrase "What time," more extreme measures are called for. How about "At exactly what time of day will we (do something)?" That ought to do it.
Student or Learner