I suffer from muscle damage and regular spasms in my lower back but after a pretty severe episode a few years ago (I tried to go back to playing football..) I now know what is manageable and when to stop.
All the regular stuff like hot baths, muscle rub, stretching, looking after your posture throughout the day, taking care when lifting etc are all beneficial and important. If the pain is getting unmanageable, I'd recommend booking into a private physio for a few sessions to let them diagnose it and work out what exercises best suit your problems. Go see your doctor too, they'll refer you for physio if they feel it necessary but it'll take maybe 3-6 months to get an NHS appointment, although I've had quicker turnarounds than this but back pain is normally long term so at least this'll put the wheels in motion.
I've been told that cycling isn't particularly good for your back, especially if your arms are taking the brunt of the weight, therefore your core is not engaged. Plus on a road bike your body is locked in a hunched over position so maybe think about riding more upright or changing to a flat bar for a while.