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This article will help: https://www.gardeningknowhow.com/ornamental/shrubs/boxwood/trimming-boxwood-bushes.htm

Boxwoods are popular for this. They grow slowly so do not need to be repotted often. They will at some time need to be repotted, but it could take several years. Regular potting soil is best. Also, make sure that you also add plenty of slow release fertilizer to your container plants. This will help with making sure that your evergreens have plenty of nutrients to keep them growing well. The addition of some mulch to the top of the container will not only help retain appropriate levels of moisture but will help to slightly acidify the soil as well, which most evergreens like. It will also be helpful during winter, along with wrapping the base with burlap to keep them warm.

I would look for a variety that is well suited to your growing zone, especially since it's going to be in a planter. In fact, one zone below would be even better. You also might try looking at Inkberry. It has a similar look to boxwood. It grows slowly as well, but is typically a little more cold hardy than boxwood and can take the winter better.

Since your shrubs are newly planted, the most important factor is water, especially once temps begin to rise. There is really no need to worry with fertilizer (and feeding plants during hot or dry weather is not a good idea), though adding a balanced fertilizer may help some. Be aware that they have only been in the ground a couple months at best and it takes time to acclimate to new surroundings. This is why growth is slowed. Pruning them would have actually helped the plants focus their energy towards root growth rather than foliage growth. Be patient. Your shrubs will eventually begin to show new growth.

Early to mid-spring is the recommended time for pruning yew shrubs, though you could do some light trimming only in the fall. Also, drastic pruning will greatly reduce its greenery, so any trimming you perform should be light unless you are willing to live with the loss of green for awhile. Yews can be cut back into older branches but new shoot emergence from old buds will take years and new growth will likely be thin and unattractive.

As for the arborvitae, they too should be pruned back in early spring prior to new growth. These plants, however, can take more drastic pruning and will regrow quickly.

Though the Boxwood like sunshine they need to be protected from the hottest part of the day as the summer heats up.
Feed and water the plants as normal.
Give them time to recover, then you can prune up the plants to remove the dead material.
Here are a few links for you.

https://www.gardeningknowhow.com/ornamental/shrubs/boxwood/boxwood-care.htm

https://www.gardeningknowhow.com/ornamental/shrubs/boxwood/yellow-or-brown-boxwoods.htm

You should purchase a good quality pot with good drainage. Check the holes in the bottom of the pot.
Buy a good quality potting soil. Do not use dirt from the garden as this will compact to hard.
I recommend filling the entire container with soil. If the pot will be heavy and need to be moved you can purchase a little pot lift with wheels.
Here are a few links for you with more information.

https://www.gardeningknowhow.com/ornamental/shrubs/shgen/potted-shrubs-growing-shrubs-in-containers.htm
https://www.gardeningknowhow.com/special/containers/container-plant-watering.htm
https://www.gardeningknowhow.com/ornamental/shrubs/boxwood/boxwood-care.htm