How can a customer change a field (e.g. the priority) after he has created a ticket?
Depends of your Jira version and permissions policy.
If you can add any transition button into customer portal, you can configure PostFunction for this transition, which will increase issue priority.
If not, there is possibility to provide speciall permissions for user group, who can increase priority directly in Issue interface.
That does not answer my problem. The customer shall have the possibility to explicitly change the priority by himself at any stage of the process ...
Cannot do this OOTB. You can look to addons. Here is a thread that offers a solution.
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