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Buy a round timer. You can get an cheap Everlast one for around $15. Then practice ‘shadow standup.’ I say shadow standup as opposed to shadow boxing, because you want to work in as many elements of MMA stand up as possible. Box, kick, sprawl, level change, shoot, etc.

Concentrate on good form with your techniques, good footwork, and with constantly moving. While no replacement for sparring or training with a partner, it is a great workout and will improve your standup.

Training for ground work with no equipment is harder, but if you can pick up an old heavybag and/or plyoball at the local thrift store for a few bucks, you can work your balance and hip movement by rolling on the plyoball and your knees, elbows, ground and pound and moving from position to position by using a heavybag laying on the ground.

I know that is not a direct answer to your question, since you said ‘no equipment’ but hopefully it will give you some ideas to get started.

Shadow kickboxing. Sprawl and bounce back to your feet. Push-ups. Crab crawls, bear crawls, knee over knee’s(replicate shots). Practice throwing combinations before going for a takedown.