I own an apartment in Copacabana and I spent a lot of time in Rio. I don't get there as often as I would like these days. I spent more time there when I was a single bachelor. You will be fine during the day. I only had one incident with an attempt of someone trying to rob me and that was probably my own fault for walking around super late at night and some kids were begging for money and I took out some money to give to one kid and then got swarmed like bees and they tried to get some cash from my wallet and my watch but I just went all Bruce Lee on them. Ha, ha. (They didn't get anything).
The biggest issue with Rio is that the slums and favelas are next to nice neighborhoods as well but honestly if you aren't wearing any jewelry you will be ok during the day. I've walked around quite a bit and never had issues. It's beautiful running along Avenida Atlantica and also in Ipanema. You can run all along the beach back and forth only in Copacabana it's a few miles so you can run back and forth or run along the beach and then over to Ipanema along the beach. You won't have issues.
I was just there in July with my family. My wife has been there many times as we would stay in my apartment but my kids just went to Rio this summer and loved it. I looked and the distance between Leblon and Museu do Amanhã is just over 8 miles.
Here is a good website where you can see some running routes:
A guide to running in Rio de Janeiro. The best places to run are along the spectacular beaches & waterfront, the Urca neighborhood, Lagoa loop, and Olympic sites tour.
greatruns.com
Be careful at night depending on where you go. I speak some Portuguese. One thing you won't have trouble finding is a new Novia. Cariocas are very friendly and easy going and also beautiful. It's funny the first time I went I thought a girl was stealing my phone but she was putting in her phone number.
Rio is a magical place. I think one of the most naturally beautiful in the world. The view from Sugar Loaf is amazing.