Is there a SIM card in Japan that provides slow (128 kb/s or so) but cheap Internet service?
Currently DoCoMo provides a ~10 USD (1080 JPY) SIM with 128 kbps data (for 20 days from first use), called the "Japan Welcome SIM": https://wow-j.com/en/sim_wifi/
You can use your smartphone at a communication speed of 128 kbps regardless of your data volume.
You can watch (tourism?) videos/ads to "earn" high-speed data. The limited-speed data is always available and unmetered. You can also pay for extra high-speed data, though the prices are a bit higher than other providers per-GB.
This has been available since late 2017 and will likely be available through 2020 at a minimum.
There's also a completely free variant, but it must be picked up in specific (less-popular?) regions. They appear to be sponsored by local companies, probably to attract tourists: https://wow-j.com/en/sim_wifi/plan0/