The sack looks like the one here on a white pine. 

Removing the bags by hand is a good control option for small trees with light infestations, but you may need to use scissors or hand pruners to cut the strong silk fibers that attach the bags to the tree. For heavy infestations or large plants, spray with an insecticide containing spinosad (Fertilome, Bonide, Monterey and Greenlight companies all sell spinosad products). For best control, apply sprays shortly after eggs have hatched in the spring and before caterpillars have grown large enough to cause significant damage.

Avoid spraying with products like carbaryl, malathion, acephate, or pyrethroid insecticides (permethrin, cyfluthrin, bifenthrin, etc.). These products will kill bagworms, but they are also prone to trigger spider mite outbreaks. https://extension.msstate.edu/content/bagworms

See also: https://extension.msstate.edu/node/7261#bagworms

https://www.canr.msu.edu/news/bagworms_becoming_more_common_in_michigan

More About Bagworms and Treating Arborvitae for Bagworms:

Bagworms are dark brown caterpillars that love to feed on over 128 different plant species. However, arborvitae, red cedar, and juniper are the most common plants that bagworms will eat. The insects are only about an inch long. They weave silk bags (cocoons) as they feed on the branches. The bags are only about 2 inches long, but the infestation can be daunting if not treated right away. The females will actually move the bags with them as they feed on different sections of the plant. The females do not leave the bag. The males are black moths that will fly to the silk bag and mate. Each female can lay up to 1,000 eggs at a time, which can overrun arborvitae in a hurry if not dealt with promptly.

For light infestations, you may be able to just pick the bags off of the tree. Work in sections and try to remove all of them. Each female left behind can produce up to 1,000 eggs, so it is important to not leave any behind. If you cannot reach the top of the tree, you may be able to use a hose to spray the bags off of the tree with a strong spray.

Once you either pick the bags or knock them off, be sure to collect them all and dispose of them. You can either burn the caterpillar and bags or drown them.

If the infestation is too much to treat manually, you can spray bacterium Bacillus thuringiensis (Bt) on the plant. It should be safe for birds and other animals, but will kill the caterpillars once they eat the arborvitae. Treat according to instructions. The best time to treat is in the morning or evening when the worms are feeding. June is usually the best month to use a spray pesticide. If all of your arborvitae are not infected, it is still a good idea to treat them all at the same time.

To Answer your question: Will The Arborvitae Recover From Bagworm Damage?:

Unfortunately, it can take a while for arborvitae to recover from bagworm damage. The brown spots may recover or may not. A good way to tell is if you use a fingernail to look for green tissue inside the branches. If there is some green, the branch may still be viable and add new growth the following year. If the branch is dead, it is usually best to just cut the branch off. Even if that means cutting out big portions of the arborvitae. When the plant does recover, it happens from growth on the tips of the branches, which will eventually cover up the damage further inside of the plant. Sometimes, this can take several years if the damage was bad enough. You may decide to just replace the plant if the damage is bad enough or makes the plant look uneven in appearance.

https://www.uky.edu/Ag/Entomology/treepestguide/arborvitae.html

This web-site has a photo of the bagworm which is almost the same as your photo.

I hope this information helps.