You're looking for a sesshin (接心), a multi-day meditation course arranged by Zen temples that lets you experience an approximation of a monk's life. However, since you (presumably) will need English instruction, these are usually only arranged a couple of times of a year; it's not really possible to rock up to a temple with no advance warning and ask them to put you up for a few days. (Well, you can always ask, I suppose, just don't expect a yes unless you've got someone to introduce you.) I'm only aware of two temples in or around Tokyo that run them regularly:
- Tōshōji (東照寺), in Shinagawa, Tokyo. 5-day sesshin twice a year.
- San'un Zendo (三雲禅堂), in Kamakura. Foreigner-friendly courses 4x/year.
See Zen in Tokyo and Wikivoyage's Meditation in Japan pages for other leads. Or head to a temple hotspot like Mount Koya, which is chock full of shukubo (mostly Shingon though, not Zen).
EDIT: You say you're looking for shukubo, but the ones I've stayed in (probably a good half-dozen?) are very much lodges first and temples second, with tennis courts, hot springs, sake served with your meals, etc. You're usually allowed to join the morning service, but that's about it, it's pretty far from a monk's life. Nonetheless, taking your request at face value:
Syukubo.com (Japanese only) knows of only two in Tokyo, and both only technically -- one's up a mountain in the far west reaches of the prefecture (Mitake-san), the other's a two-day boat ride south (Ogasawara Islands). This place in Chiba looks otherwise OK, they'll even let you work for free if you pay the ¥8500 a night, but they're Nichiren and thus the polar opposite of Zen. Most of the rest seem either closed, or tilted way towards the resort-y end of the spectrum... but good luck hunting!