Two possible solutions:
- If you are connected to a Google account, your contacts are probably uploaded to the cloud. You can find out by logging in to your account via a computer.
- Go to the contacts app and turn your phone to landscape mode (90° rotation). If you can see the “three dots” button at the top of the screen, click it. Then choose “Import/Export”.
If none of those work, try this little tutorial I wrote a while back. It will make it possible for you to turn the screen upside down and use the menu button on the working part of your screen.
Edit: Here a screenshot about what the 2nd option would look like:
