Quarantine comes first
Japan's Animal Quarantine Service requires advance notification, up-to-date rabies vaccination records, and in many cases a waiting period before a dog or cat can enter the country without an extended quarantine stay at the airport. These requirements are detailed, country-of-origin dependent, and genuinely need to be sorted months ahead — this is worth handling directly through Japan's official animal quarantine guidance rather than a general travel guide.
Finding pet-friendly accommodation
Once the quarantine side is sorted, a smaller subset of hotels across Osaka explicitly allow pets, generally concentrated among mid-range and business hotel chains rather than at the high end, and almost always requiring the pet to stay in a carrier in shared spaces like lobbies and lifts.
Getting around with a pet
Osaka's subway and trains generally require pets to travel in an enclosed carrier rather than on a leash, and most restaurants and shops don't allow pets inside at all beyond a carrier — worth factoring into how much of a day out is realistic with an animal in tow.
If a pet is core to your trip, start on the quarantine paperwork as early as you can — it's the single biggest planning item, and it's not something that can be rushed close to a travel date.