Skiing / snowboarding can't really be truly budget. Give you an example, my last year ski trip to nozawa onsen:

"Skis, boots, poles, and clothing (jacket + pants) rental cost ¥8,550 for two days; Ski Lift ticket cost ¥9,500 for two days."

On top of that, my accommodation was ¥14000/night for a small japanese style inn that didn't even have own bathroom (shared toilet shared bath shared sink area).

Anyway Gala Yuzawa is close to Tokyo and is a popular nearby place to hit the slopes.

If it's your first time, either bring someone along who can teach you the basics, or seriously consider signing up for at least half a day of lessons.