Yes - Apple Watch Series 1 can be used to track an outdoor running workout without taking your iPhone along.
To record your workout as a specific activity, be sure to track it using the Workout app (or a third-party app that shares data to Health), selecting Outdoor Run as the workout type:
When used without taking the paired iPhone along, in order to estimate distance, the Workout app relies on data from the built-in accelerometer. Be sure to allow the arm on which you are wearing your Apple Watch to swing naturally whilst running.
After you have completed your workout, once your Apple Watch and iPhone have reconnected via Bluetooth, your workout results will be synced automatically to the companion Activity app (and Health app) on your iPhone. The results can then be reviewed on your iPhone at any time:
For improved accuracy of estimations, you may wish to calibrate your Apple Watch by taking your iPhone along during some initial workouts. Detailed instructions for doing so are provided here:
Calibrating your Apple Watch Series 1 or earlier for improved Workout and Activity accuracy - Apple Support
Other tips for getting the most accurate measurements are detailed here:
Get the most accurate measurements using your Apple Watch - Apple Support