What's the name for period of arrival ahead of time?

Given your need for a single word for UX purposes, as explained in your comment, I suggest ahead or early as your best options. Your goal is to provide information that will be understood quickly, with a minimum of additional information. While someone might come up with a word that's better as a true antonym of delay, it doesn't do you much good if the intended audience needs to be told what it means.

For a very pertinent example of this principle in action, see this tool, where a bus can have a 5 min delay or can be 5 mins early.

Common Usage

“[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”

Edit:

“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.

Technical Usage

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