A post ban will never be lifted automatically. Not without the user actively working on improving their content, and it will not just "go away".
From the Help Center page:
The only way to end a posting block is to positively contribute to the site; automatic bans never expire or "time out". Begin by fixing your existing posts.
Taking into account the user's activity on other sites on the network is something I would be opposed to. You can be a great participator on Cooking but might have completely missed the scope of Super User. Contributing to another site is not related in any way to your actions here.
Or if he is not an expert to start with will he be doomed to be a newbie for life without the opportunity to improve?
This is not a question of expertise. Remember that this is a Q&A site. Some people have questions, some have answers. Some have both. Not all are experts—and someone's got to ask the questions, right? What we need is good questions.
You can be a total newcomer to a topic and still ask great questions. The quality of your posts and your experience in a topic, those are two completely orthogonal concepts. Remember that we are all hear to learn, to some extent. Nobody's born an expert on a topic, but asking good questions is a great way to get there.
Improvement with regard to post bans is always measured by the feedback you get from the community. Upvotes positively contribute to your account's standing, so does posting good and relevant answers (even when you can't ask questions). All you need to do is create good content, and improve the bad content you have, as long as it's salvageable.
Now, regarding your account's standing, we can help you undelete a few posts you've deleted yourself. Having too many deleted questions is never good, as you can imagine. This, however, is as much as we can do for you. Let us know if you'd like to go that route.