Verb conjugation of "despuntar" in Spanish

I came on time and two hours later I arrived at Tuxla Gutieres. I was always suspicious here and I never knew whom to trust. Somehow I got to the airport a few hours too early but this is better.
After placing my luggage there I decided to walk into town and rent a bike. The neighbor of the guest house stopped me on the street and asked me if I was Mike. She told me that she was Luisa.
We told the cabdriver the streetname we would like to go and he brought us there. This time the place did look like a three star hotel and we booked a night. I went to pay for the taxi ride.