Is it OK for SEO to keep same URLs on new site without redirecting even though the site was down for a while?

What you have asked is a bit unclear with respect of if the 404 errors were occurring on the new site (not great) or old site (irrelevant)

Long term thie latter should not be a major issue for your site, but short term it may reduce the number if pages Google returns and, I suspect, could lower the ranking (this is just an educated guess)