Cross-training with different sports and weights is good, because no sport works every muscle equally. Cross-training helps balance out your muscles and avoid injuries, plus it strengthens your "weak link" in your primary sport and makes you stronger in that too.

The important thing with any training (whether multi-sport or single) is that every day can't be a hard, all-out day. Don't expect to ride your hardest the day after your leg strength day. Take at least one day a week of active rest (walking, EASY cycling, Pilates, yoga without a lot of strenuous standing poses, etc.) - depending on your condition and age, you may need more than one day. Remember that working out stimulates your muscles to get stronger, but it's during your recovery time that you're actually becoming stronger.

An hour a day of running is pretty ambitious for someone who's only been working out for a month. Start with a mile. If that feels good and you don't have an excessive amount of delayed onset muscle soreness the next day, then increase your distance in small increments until you feel like you're challenging yourself.

Any time you have delayed onset muscle soreness for more than a couple of days, IMO it means you overdid it (I know some people would disagree with me, that's just my opinion). That's okay for a big event, not so much as part of your regular training routine. It doesn't mean don't train, it means take the intensity or duration down a notch until you find a level that challenges you some without making you so sore that it interferes with your progress. Once you've been strength training for a while, that won't happen, but it's easy to do early on (and any time you add a new sport).

Join in the weekly running thread for inspiration. It's a nice group of runners at an enormous range of levels.

And... welcome to TE!