Well, you'll hit some traffic if you're driving on a weekday into the city and you depart between 6:00 and 9:00 and return between 3:00 and 7:00 so figure that in.
Mulholland is an interesting experience with some great views but the road is quite curvy so you don't want to be driving too much of it. I would go to Mulholland first (you'll be 'fresh' in the morning and the sun will be high in the sky instead of in your face as you drive those curves).
When you arrive from Newport, you can come into L.A. on the 405 freeway (on the westside) or on the 110 freeway (into downtown, which is near the eastside of our city). So, what route you take for mulholland will depend on which way you enter L.A. (and this, in turn, might be related to which other attractions you'll be visiting on your way in and on your way out).
Spending a single day in L,A. (including a drive from Newport) will be challenging -- you can't see a lot of attractions, just a handful.
Here's a route but there are many more:
Entering L.A. via the Harbor Freeway. Exit into downtown, park and explore our downtown area. Downtown is often ignored by visitors since it's at the eastern edge of our city and is probably not as well-promoted as the westside, Hollywood, Beverly Hills (which is actually not in L.A, of course), etc.
Lots to see in downtown, e.g the Catholic Cathedral, Riordan Library, Bradbury Building, Chinatown, City Hall, MOCA, etc, etc. Exit downtown via the Hollywood Freeway, going north (the only direction you can go).
Before I forget, here's a list of the scenic overlooks on Mulholland for the section you'll be visiting while in L.A.:
and some info about the street
Okay, here's a possible route from downtown (I've marked an arbitrary starting point in downtown) to Rodeo Drive via the eastern portion of Mulholland Drive:
The stops along the route are 'scenic overlooks'. Stop at as few or as many as you please. If the curvy driving becomes 'too much', you can turn left off of Mulholland on Laurel Canyon Bl (which is also very curvy) and head 'down the hill' into West Hollywood. Turn right at Sunset Bl and drive through the famous Sunset Strip and into Beverly Hills. Turn left onto Beverly Drive (as you'd do later on, on this route) or Rodeo Drive and head into downtown Beverly Hills.
As far as what other sites to check out, it really depends on what your interests are. If you like small to medium-sized art museums with interesting collections, I'd recommend the Hammer Museum in the Westwood neighborhood, just a few minutes west of Beverly Hills. If you don't want to fuss with eating in Beverly Hills, the Napa Valley Grille is adjacent to the Museum so you can park in the museum's parking structure and walk to the restaurant for lunch.
We have beach communities, historic sites, interesting architecture, ethnic neighborhoods, etc, etc, etc so tell us more about the types of things that interest you and we can make recs.
For the trip back to Newport, head a couple of minutes west of the Hammer along Wilshire Bl and you'll see signs for the 405 freeway south just ahead. Have fun!