Your body will easily adapt to both. There is no reason for your muscles to get weaker, then stronger again, as you alternate between running and biking day-to-day, as long as you build both up gradually. What is important is getting enough rest and getting proper nutrition (protein, if you're building these muscles up).
Depending on training load, you may feel fatigue going from a hard run one day to biking the next day, or vice-versa. This is not to be confused with "muscle readjusting" and is just basically, they're tired. Having a rest day in there helps depending on how hard your effort is.
I started off with biking and built up a fairly large base, before I started running somewhat seriously. Running is a lot more impact so you have to be careful to take the time to build up your running base very gradually (I kept getting injured the first few months until I worked out my running form and eased in the miles, now I'm largely injury free running). This year I've been able to put in 35-45 miles per week average running, while doing a long ride every weekend (80-100 miles) as well as 3 double centuries squeezed in. I don't recommend the once a week long bike ride unless you've already got a large bike base built up. Alternating short rides and short runs and slowly working up the distance is probably more beneficial, if you're trying to build up both at the same time.
The nice thing is that whenever I have gotten injured running, I can still bike, and whenever I get injured biking (which is rare, most of the time I just get sore, usually my butt, from doing long rides) I can still run.
And yes, running isn't bad for the joints, unless that person has really crappy running form, already has bad joints, or does extreme distances (overuse injury). I do believe however that it is harder to start up than biking, since it has more impact, so start young and "cruise" into old age with a big base built up, and you can keep fit running or biking, whatever you desire.