“[running] ahead [of schedule]” is the usual usage.
- That bus is five minutes ahead.
- That bus is five minutes ahead of schedule.
- That bus is running five minutes ahead.
- That bus is running five minutes ahead of schedule.
I don't believe there's a common noun for this meaning, however.
One might refer to that as being “hasty,” or “rushed”
- That bus was hasty by five minutes.
- That bus was rushed by five minutes.
Neither is a noun in the same structure as you requested, though, and the connotations are bit different:
- “hasty” implies that an early departure was due to carelessness, or perhaps foolishness
- it may be more familiar to speakers in the UK* and the American South and perhaps Midwest; I believe its usage is considered archaic or quaint in the Northeast and West;
- *edit: It seems that this may not be colloquial in the UK, either; this usage might be limited to the US South. It's also a bit unusual to attribute a specific time interval to hastiness; it's generally used as an non-quantified attribute (“The bus's departure was hasty” or “the bus departed hastily” more often)
- “rushed” implies external events conspired to force the early departure, which is a bit more similar to “delay”
“Premature“ also suggests itself, but still, not a noun.
A premature departure might imply that the bus's departure caused difficulty to someone more than “hasty” or “rushed,” but it does sound a bit formal.
In technical terms, I would expect to see “advance” used as a noun, when time is understood:
- The timing pulses arrived with 10ms delay
- The timing pulses arrived with 20ms advance
… although it requires a context to understand that the “advance” is a time interval.
This is sometimes used in electronics/communications jargon.
This is similar to the usage of “timing advance” by automobile mechanics, but in that case refers to the timing as a relative position of the crankshaft rather than a time-unit (the actual time-interval this represents dependant upon the rotation speed of the engine)
- The timing has a 5° advance
I wouldn't expect to see it in use in regards to transit, however:
✗ The bus departed with 5 minutes of advance
… seems very awkward