For a first trip to LCY and with hand luggage only, I'd aim to arrive at the airport c.45 minutes before departure. Allows a little wriggle room for emergencies (though not much). LCY is pretty quick through security, even at peak times, and once through security you can be at most gates within 5 minutes (less than that for some gates).
There's not much to do or anywhere really to sit after security; you can spend 30 minutes having a coffee or something but really would not want much longer.