Cfrtjryk

Example: Don't be angry at me for asking this question, but are you rich?

Is there a term for the sentence in bold?

Preamble is accurate; however, I believe in casual conversation that preface would be more usual.

preface noun 2 : the introductory remarks of a speaker or author (Merriam-Webster)

It is often used in its verb form for just this sort of hedge (also a good option):

Let me preface this by saying...

(This Google Search, and a broader search with lots more examples)

So in your example, you could say

The speaker included a conciliatory preface before the question.

or

The speaker prefaced the question with a plea to not take offense

If you were the speaker, you could even say

Let me preface this by asking that you not get angry. Are you rich?

or

Let me preface this by saying that I hope I don't offend you, but are you rich?

Edited to add:

As EdwinAshworth points out, while preface-as-a-verb might be as likely to refer to a casual or spoken statement as to a formal, written work, the primary meaning of the noun is still an introduction to a book. However, I do believe the noun is used this way, perhaps influenced by the idiomatic usage of the verb. Some examples from around the internet (all emphases mine):

I have noticed in the past few years that many of my students tend to
blurt out, “Wait” as a preface to a question they have. (Jeff
Ortman, blog post)

A preface to a question that announces its content softens the
harshness or abruptness of the question itself. (Carol E. Westby,
Ethnographic Interviewing: Asking the Right Questions to the Right People in the Right Ways)

How come people say "no offense" as a preface to something they know darn well is offensive? (Discussion board posting)

"Bless her heart" > A phrase used as a polite way of pitying a person or as a preface to
an insult (supposedly to lessen the injury). (Cindy Pierce, AOL
News, "Dallas Slang")

Everyone understands that ‘with all due respect’ is usually the
preface to an insult (Comment on a blog)

And used sort of adjectively

I just want you to know that this blog post will be about verbal
tee-ups, the preface-y phrases—“I’d like to say,” “To be perfectly
honest,” “No offense, but,” “I hate to be the one to tell you this,
but”—that either drive you crazy or allow you to get away with
conversational murder. (Katy Waldman, Slate, "No Offense, but
Verbal Tee-Ups Aren't Actually Polite")

In finding these examples, I noticed that prelude is also often used, particularly in the set-phrase "prelude to an insult" (this has even been the title of a book and a song). Preface still seems more common outside of that particular usage, and a little more neutral in terms of connotations (whereas preamble feels legal and stuffy due to its association with things like the US Constitution, and prelude feels elegant and possibly snooty due to its musical meaning, either of which might be desirable depending on usage).

In short, it doesn't appear that there's a single agreed-upon term, though several would be widely-understood. You could also call it something like a "preemptive pacification" which would be fairly precise but isn't particularly familiar.

It's called a preamble.

Preamble noun
1 A preliminary or preparatory statement; an introduction: he could tell that what she said was by way of a preamble [mass noun]: I gave him the bad news without preamble
- ODO

Here are some examples taken from the internet:

• As a preamble to comments about the department's policy on distinction, please keep in mind that distinction in comps does not really matter much when it comes to your future plans. - Chemistry Comps 2015-2016




• Nancy Leigh's words would, no doubt, serve as a preamble to a major showdown between the two sisters ... - Welcome to Higby by Mark Dunn

There's been some discussion on whether the leading part of the question's example sentence should be called a preamble or a preface. Both are valid. The term preface has a slightly stronger connotation of leading into the subject, while a preamble has a slightly stronger sense of simply being the portion before the main text. The portion in bold prefaces the question "Are you rich?", and serves as its preamble.

Such remarks might be described as preliminary as they're delivered before the main subject is broached.

If they come at the start of a conversation or change in subject then they might be described as introductory, as they set a frame for the information that follows.

They might even be described as cursory as they're often brief or hasty so as not to distract from or delay the main thought.

You might call this kind of phrase a lead-in as it's intended to guide (i.e. "lead") the listener toward or along a certain line of thought as the main subject is approached (i.e. as you're "getting into" the main subject).

Similarly, it may even be called a warmup if it's intended to make a sensitive subject or information less offensive. This alludes to "warming" the listener up to an otherwise "cold" or "chilling" topic.

Let us acknowledge, then, that we have a preamble.
But if it's intended to guide the listener toward or along a certain line of thought,
Then among all preambles, is there a preamble?

A certain maxim of Logic which I have called Pragmatism has recommended itself to me for divers reasons and on sundry considerations.

Having taken it as my guide in most of my thought, I find that as the years of my knowledge of it lengthen, my sense of the importance of it presses upon me more and more.

If it is only true, it is certainly a wonderfully efficient instrument.

It is not to philosophy only that it is applicable.
I have found it of signal service in every branch of science that I have studied.

My want of skill in practical affairs does not prevent me from perceiving the advantage of being well imbued with pragmatism in the conduct of life.

This great hope is embodied in the conception of truth and reality.