This time tomorrow, we'll be in the middle of climbing Mt. Fuji.
Literal
Tomorrow [possessive-の] this-about-time, we [topic-は] Mt.-Fuji mid-climbing [polite-でしょう].
~中 attached to a Sino-Japanese noun is a productive suffix meaning 'in the middle of / during' — 工事中 ('under construction'), 食事中 ('eating'), 登山中 ('climbing a mountain'). It's a compact way to express 'doing X' when ~ている would be bulkier.