I just find Persisto. Can I use this with Cordova?
You can use it with Cordova, but it appears to just be a wrapper around localStorage, so is no better/more persistent that using directly using localStorage.
If your Cordova app is targeting iOS then you should be aware that any locally persisted data stored inside the Webview (localStorage, WebSQL, IndexedDB) is considered cache data and therefore can be wiped at any time if the device runs low on storage space.
Also I'm looking for a synchronous library. Do you have any advice?
TL;DR: I don't think you'll find a sychronous interface to a better storage mechanism because the more durable/reliable mechanisms are all inherently asynchronous.
Ultimately it would be better to bite the bullet and rework your code to work with an asynchronous storage mechanism.
For reliable persistent storage that isn't going to get wiped by iOS at a whim, you could use a native SQLite DB via the cordova-sqlite-storage plugin.
For an simple interface to it, you can use a wrapper such as localForage with the cordovaSQLiteDriver adapter.
Or if your content is more file-based, you can store it using cordova-plugin-file to a durable storage location.