This really depends on what you want to do with this "real time" after it's converted, but I'm going to operate on the assumption that you just want formatted output from the [suncalc] function you're using for readability.
If you just want to display the time in 24-hour / 12-hour format as a string, you could append either of these pairs of lines to the end of your [suncalc] function just before the 'return' statement:
# Use these two lines for 24-hour format
sunrise <- format(as.POSIXct(sunrise*3600, origin="2001-01-01", "GMT"), "%H:%M")
sunset <- format(as.POSIXct(sunset*3600, origin="2001-01-01", "GMT"), "%H:%M")
# Use these two lines for 12-hour format (AM/PM)
sunrise <- format(as.POSIXct(sunrise*3600, origin="2001-01-01", "GMT"), "%I:%M %p")
sunset <- format(as.POSIXct(sunset*3600, origin="2001-01-01", "GMT"), "%I:%M %p")