I recommend reading Japan Mobile Tech's (formerly Softbanksucks)
General information for smartphone users page, specifically the Softbank section, for information on how to get the One X (or any other smartphone, for that matter) working.
As mentioned above, Softbank is offering a la carte sims for BYO phones, but a) as a requirement, the phone must display the Japanese certification mark either stamped or stuck to the phone, or on the display (lightning bolt above a 〒 inside a circle), and b) they are not offering iPhone (read, micro) sims.
This means you basically have to follow the procedure outlined in the JMT blog post above, and cut the sim down to size to fit in your phone.
The procedure, generally, is to start a service with any phone on Softbank, and subscribe to the S! Basic Pack (¥300/month + 5% tax). Subscribe to the Unlimited Packet Discount S plan, and then change your data plan to the Unlimited Packet Discount for Smartphone plan (¥1029 - ¥5985/month).
You can do this either by getting the customer service rep to do it (although they will wonder why you want the Smartphone plan when you are signing up for a regular phone, and it's not a good idea to tell them why you want to do it), or you can do it yourself on my softbank site. The steps are outlined in the excellent JMT post above, so read them carefully. Finally, you can swap the SIM from your Softbank phone to your smartphone once you have confirmed that the proper data plan is active.
After that, to get MMS working with your @softbank.ne.jp address, you will need to make sure that the APN settings are correct (open.softbank), and then download a custom Mms.apk which has the UserAgent that Softbank expects (otherwise all MMS sending and receiving is blocked). My ICS .apk can be found
here, and for Gingerbread and below, Takuo posted some excellent .apks on his blog
here. These Mms.apk files replace the stock Mms application (and use the same database, so if you simply flash over and reboot, your messages should still be there - take a backup just in case). You will need to adb push them to /system/app/Mms.apk and reboot to be able to use them, so a rooted device is a must. If you can't (or won't) root, you can check out Softbank's official S!Mail app from the Play store, but I don't believe it supports ICS at this time. Good luck!