That route from Arches to the Grand Canyon is a relatively easy drive. Decent roads without a lot of traffic until you get close to the GC. It's only about 325 miles and would take you roughly 6 hours to drive. Of course, there are a lot of cool things to stop and look at so your departure/arrival times will probably be more than 6 hours apart. Add to that, once you get onto Desertview Drive, along the South Rim, you will want to stop at all the different lookout points (the first one will be Watchtower). This will also add some time.

Of course, you could go back and do those lookouts the next day if it's late and you're tired. But, even so, driving once you get into the national park will be a bit slower so plan for that.

There's no need to go to Flagstaff. If you leave directly from the Grand Canyon Village area, you can just go straight south to Williams, a bit west of Flagstaff, to pick up I-40.

I've driven US89 and found it a very easy road to drive. I thought it was relatively quick. I think you could easily see both Bryce and Zion before getting onto I-15 and heading toward home.

If you have time, have you considered visiting Arches, then Bryce, then Zion, and then driving around to the Grand Canyon? Your route would look something like this. Of course, whether this works depends on how many days you squeeze out. You would want at least a half-day for Arches and Bryce, a longer day for Zion, and a full day for the Grand Canyon. A decent itinerary might look something like this:

Day 1: Visit Arches in the morning to early afternoon and then drive the 275 miles to Bryce in the afternoon/early evening.

Day 2: Visit Bryce in the morning, If you just do the look-outs to see the amazing views and rock formations, you could probably be done by Noon, and then drive the short 86 miles to Zion, exploring it for the rest of the afternoon and evening.

Day 3: Drive the 250 miles to the Grand Canyon Village. If you get an early start, you could even squeeze in time to go visit the North Rim! It's very cool and worth doing. You could still get into the South Rim area by late afternoon and enjoy a few lookouts and watch the sunset.

Day 4: Get up early to enjoy the morning light at the GC...sunrise if you can crawl out of bed in time is great! See what you haven't seen yet, and you could probably be on your way to Williams by early afternoon.

Of course, these timeframes are assuming that you're more into casual walks and not into hiking. If you want to take time to do some more hiking into the various areas, this timeframe might be too tight for you.

Anyway, just a thought, if you can squeeze out the time.