An Opportunity needs to have an amount rolled up from a custom object that is...

Write a trigger on the Opportunity object and use tree sorting to sum the amount for all related child objects under the Opportunity.

Use the Streaming API to create real-time roll-up summaries.

Write a trigger on the child object and use an aggregate function to sum the amount for all related child objects under the Opportunity.

Use the Metadata API to create real-time roll-up summaries.