Drain the tofu and place it between several layers of paper towels, or between a thick kitchen towel. Place on a rimmed plate and press down with a heavy object, such as a skillet. Press for 30 minutes.
In the meantime, bring a pot of water to boil.
Add the soba noodles and cook until just barely al dente, about 2 ½ to 3 minutes.
Drain the soba noodles and rinse with warm water to remove the starches.
Divide the soba noodles into bowls.
To make the broth, add 8 cups water and instant dashi to a pot. Bring to a simmer.
Season the broth with soy sauce, mirin, and sesame oil.
Dice the drained and pressed tofu into 1-inch squares.
Add the tofu and bok choy to the broth and cook until the bok choy is bright green and soft, about 5 minutes.
Ladle the broth, tofu, and bok choy over the noodles.