That really depends on a few other variables.
Shared Preferences are great for iterable key/value pairs, but aren't really designed for relational data. They're also meant to store a finite amount of data, not an arbitrarily large dataset. For instance, you would save a user's name in SharedPref, but probably not a complete history of every score the user achieved in every level, or the definition of every level.
SQLite is much better at relational data, since it's a relational database, but it might be overkill if you only ever want to hold on to one object at a time. Typically you'd need to set up a content provider, but I assume ORMLite would handle that for you.
So: If you're only storing one or two instances of this json object at any given time, probably shared pref just for convenience. If you're storing arbitrarily large number of them, and the data it relates to, SQLite database.