does pbn works for deranked website?

i have a deranked website which got deranked after i forget to renew the domain. my question is can i boost it rank buying pbn? or what type of link building should i go for boosting the deranked website on serp?

It depends on how much ranking you lost and niche but in general pbns should work if you get good quality ones or good quality guest posts could boost your ranking or niche edit it really depends on what your goals or cost you are willing to pay to get it's ranking back.

Site went offline for how long?

Depending on time, you can rank it again easily depending on how much competition changed for those keywords.