You do not ride to Solovetsky so much as you get yourself and a bike onto a boat. The archipelago sits in the White Sea, on the southern edge of the Arctic, and the ferry or the small plane from the mainland near Kem is the only way out. Once there, the island is roughly twenty-five kilometres long and has a network of gravel and dirt tracks that make it a quiet adventure ride rather than a road trip. Allow for rain, for wind off the water, and for very long summer light around midsummer when the nights hardly darken.
The reason people make the effort is the monastery. Two monks, Savvaty and Zosima, founded a hermitage here in the fifteenth century, and it grew into one of the richest and most heavily fortified religious houses in Russia, with walls of huge granite boulders that you can see from the sea long before you land. In the 1920s the Soviets turned the whole thing into a camp, one of the first of the Gulag system, and thousands of prisoners died here. Both histories sit on the same ground and nobody on the island pretends otherwise. The monastery has been returned to the church and is again lived in, which gives the place an odd weight, tender and grim at once.
Beyond the walls the island is mostly lakes, spruce and pine forest and low hills, linked by canals the monks dug long before engines existed. There is a botanical garden planted by the monks that has no business surviving this far north, and the coast has white beluga whales in summer, if you are lucky with the boat trips. Small Mucksalma Island is reached by a causeway of stones roughly a kilometre long, built by monks, and it is an easy ride or walk across the shallows, with seabirds and a view back to the monastery.
Practical matters are thin. The island has a settlement with shops and a couple of places to eat, but supplies come by boat, so fill the tank on the mainland and bring what you cannot do without. Tracks turn to mud in rain, mosquitoes are fierce in July, and the realistic season is June to early September. Check current rules for travel to Russia before planning anything.