Aquarium and museums
Kaiyukan Aquarium is the obvious full-morning or afternoon option, genuinely built as an indoor experience rather than something diminished by weather. Nakanoshima's museum cluster — the National Museum of Art, the Nakanoshima Museum of Art — and the Osaka Museum of Housing and Living along Tenjinbashisuji all work just as well, if not better, with rain outside.
Spa World and Den Den Town
Spa World's themed bathing floors are entirely indoors and turn a grey day into a genuinely good excuse for a long soak. Den Den Town's multi-floor electronics, anime and retro game shops are similarly weatherproof, and dense enough to fill several hours without stepping back outside for long.
Underground Osaka
Umeda's sprawling underground mall network beneath and around Osaka Station connects shopping, restaurants and department store basements (depachika) without much need to surface, and it's worth exploring on its own terms even beyond its usefulness as weather shelter.
A family-friendly indoor option
For travelling with children, LEGOLAND Discovery Center in Kyobashi is a smaller, fully indoor alternative to a full USJ day, sized closer to a half-day than a whole one — see the wider guide to Osaka with kids for more family-specific planning.