Is it possible to cut Russia from the internet?

The masses, yes. Metaphorically cut the cables heading east and null route Russian IP allocations. It would kill the vast majority of their traffic.

The government and security organizations? I'm sure they've contemplated this scenario and have multiple redundant links with some bandwidth via third-party nations. Will India, China, Pakistan, etc. all cut them off too? I'm reminded of how even the North Korean leadership has Internet access via China.

I wonder where Russian undersea cable cutting subs are now. Retaliating by severing US from the UK seems real and will cost more than turning off natural gas. Norway reported a cut earlier this year, though who is responsible is still unclear.

Not very smart move.