Adverb for the verb spend [duplicate]

Solution 1:

No, "inactively" doesn't work here. This is what the relative clause says:

Young people spend X amount of time inactive.

Which could be read along these lines:

Young people are inactive for X hours a day on average.

The adjective "inactive" says something about the subject "young people".

Solution 2:

No. The adjective is correct.

"the amount of time that young people spend inactive."

means "the amount of time that young people spend as a result of being inactive."

This use of adjectives has two categories:

Resultative: https://www.usingenglish.com/glossary/resultative-adjective.html

I shot him dead = I shot him [and, as a result, he was] dead.

He beat the metal flat = He beat the metal [and, as a result, it was] flat.

Descriptive:

She arrived at the meeting drunk = She arrived at the meeting [and she was] drunk.

But

She arrived at the meeting drunkly = She arrived at the meeting [as if she were] drunk. Here she might not have been drunk - she might have been ill or injured.

There is an interesting and readable paper on this at http://wvw.broccias.net/research/SLE2001.pdf entitled "Unsubcategorized objects in English resultative constructions" by Cristiano Broccias (Università di Pavia)