Can I get my data back after restarting my phone without having backed anything up?

In general if you don't have a backup then it's not possible.

But see this Apple article: Backup methods for iPhone, iPad, and iPod touch

Backup methods for iPhone, iPad, and iPod touch - Apple Support

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iCloud backups don't include:

• Data that's already stored in iCloud, such as Contacts, Calendars, Notes, iCloud Photos, iMessages, Voice Memos, text (SMS) and multimedia (MMS) messages, and Health data

👉 That is, if you turn iCloud -> ON you get 5 GB of free iCloud storage.

You must use the same Apple ID.

See -> How to use iCloud on iPhone -> How to use iCloud on iPhone – Apple Support (UK)