You can tailor your workout to benefit cycling. As for time, find your priorities - if you really want it you can make it happen. It may not be easy but that's part of the battle. When I w as younger I had 3 jobs which gave me 80 hours of work. I'd lift nights after work for 2 or 3 hours and sleep 3-4 hours. I ate very carefully and well. I was about 270 lbs, my bench maxed at 240 lbs which is ok but for being 5'5 I was rather happy with it. I tracked my progress and ended up maxing my stand still to full sprint speed at 26.8 mph. Again, not huge but enough to keep me happy. I did 2 days a week for each muscle group, I divided into arms, abs and legs and I always took a complete day off as a rest day.

Weight training may not seem important but it makes all the difference on the bike. Being able to recover faster and exert more power faster were two areas which i saw a noticeable improvement. I also gained a little more coordination and balance from using free weights.

I read a study that suggested incorporating cardiovascular into your strength routine could reduce the chance of a heart attack by 30%. It makes sense. No use in lifting weights if your body(including your heart) can't handle it. If you decide to get serious about lifting I'd consult a doctor about your heart health, there is a condition where some people's bodies can develop too much muscle around the heart and the end result usually isn't good. (I normally wouldn't mention this but I've lost two amazing, great friends to this - One friend passed two days before my birthday one year).