I’m 48. I find that if I do yoga first thing in the morning, it is of the most benefit to me. It loosens me up, gets the energy flowing and my day is soooo much better. I feel more awake, more energetic, than on days that I don’t do morning yoga. Those days I may have stiffness and soreness or just general lack of energy throughout the day.
I too am not a morning person. I take at least a half hour to an hour to really wake up. If I spend even 10 minutes doing yoga (some meditation and some asana) it makes a huge difference in the rest of my day.
I also like to do some stretching in the evening before bed, as it helps me relax and sleep better. You will also eventually find yourself feeling less stiff in the mornings. 
My recommendation is to do 20-30 minutes of yoga in the morning, or even a full hour, whichever works best for you. Then do yoga again later in the day.