Is it possible to display a 'shop by price' menu?

Not automatically. However, you could create one category called "prices" which would be unpublished and containing these three price categories beneath. Then, for each product, you could add one of these three categories.
Then, you could create a new module of the type hikashop content module via the joomla module manager and set it display the categories beneath the prices category.

Upon clicking on one of the price category, the user would then see a listing of the products in that price category.

It's in the todo list already.

Our schedule is already full for the next month with other things. We don't have any predefined roadmap after that but filters is not THE priority so it won't come before a few months I would say.