So your Wordpress host doesn’t allow custom themes? Because theoretically, you should be able to make a ‘modern’ theme that creates the same HTML + CSS as your beloved vintage theme.
I sympathise with wanting to keep the presentation. It was made in and for this context. It’s like reading an old magazine, you get so many extra clues about the context from the design. If a magazine does a redesign, it only affects the issues going forward. In my ideal world, blog redesigns work the same way.
I’ve done my share of maintaining and archiving older websites, where I tried to keep the design in place while updating the underlying infrastructure (or converting it to static HTML). The one thing I would sometimes change is to ‘zoom in’, multiplying all pixels in the CSS by a small amount, because the web moved to larger text sizes and screens, so older websites tend to look tiny.