Account for the relationship as it is now
Use a tone that fits the present relationship, not the closeness, authority, or access that existed in the past. For a message about how to apologize for a party that disturbed neighbors, use this factual base: state the date, place, property, noise, pet, or shared-space impact without arguing about the neighbor's character.
Ask without assigning an obligation
Explain what you hope for while making decline, delay, or a different format genuinely acceptable. Offer cleanup, replacement, access, a schedule, or another practical remedy you can complete. Make the immediate next step clear enough that the recipient can answer without reconstructing the entire history.
Keep the ending honest
Do not threaten, shame the neighbor publicly, or involve the whole building in a private issue. Use property management, local authorities, or emergency services for genuine danger rather than confronting someone unsafely. Avoid future promises, guilt, flattery, or nostalgia that imply the recipient owes you access.