You can't properly insulate it the way it is. Cellulose or fibereglass will absorb water, spray foam might work, but you couldn't repair the shingles without destroying the insulation.

It's a small area. For the simplest approach, strip off the shingles and furring, put on new sheathing, housewrap or lapped #30 felt and furring and reshingle. Inside, get rid of the cardboard, use fiberglass batts or cellulose and drywall.

I like the idea of spray foam, but to prevent real moisture problems, add 1/2" rigid foam board cut into strips between the studs and situated tight to the nailing strips. The insulation co can then spray their product right over the rigid foam. This way some air space is retained under the shingles so they can dry out after downpours, and repairs are enabled for the future; the siding can be taken off without any disturbance to the expensive sprayed foam.
Casey

Unfortunately, this approach creates a rainscreen wall that is extremely vulnerable to water penetration rather than water shedding and diversion. Which is contrary to the reason for having a rainscreen wall in the first place.

Here is a link that might be useful: rainscreen wall

Huh? It's no more susceptible to water penetration that way than it is now. There's still an air gap under the shingles, only now it only vents back outside. They are wood shingles, like a roof, they shed water. The rainscreen idea is applied to deal with horrible siding choices like vinyl and hardi which let in water at the seams (and lest I fail to mention, windows with nail fins as attachments instead of wood trim) instead of excluding water like more traditional types; wood clapboards, shakes, shingles.
The rainscreen concept didn't exist in residential 30 years ago because it was unnecessary with traditional techniques and materials.
Casey

Well, I agree, the the most thorough method would be to remove the shake, sheath, then reshingle. Unfortunately that's not an option due to the additional expense. The existing shingles wouldn't survive, so no hope of reuse there.

The shingles do shed water, but are 80+ years old, so water getting between into any cracked shingles is why I fear for insulation there. Formulating Plan B. Thanks both you've given me some ideas to work with.