I spent quite a bit of time also looking to launch a simple Python program at 01:00. For some reason, I couldn't get cron to launch it and APScheduler seemed rather complex for something that should be simple. Schedule (https://pypi.python.org/pypi/schedule) seemed about right.
You will have to install their Python library:
pip install schedule
This is modified from their sample program:
import schedule
import time
def job(t):
print "I'm working...", t
return
schedule.every().day.at("01:00").do(job,'It is 01:00')
while True:
schedule.run_pending()
time.sleep(60) # wait one minute
You will need to put your own function in place of job and run it with nohup, e.g.:
nohup python2.7 MyScheduledProgram.py &
Don't forget to start it again if you reboot.