「将棋がお強いと伺いましたが」「いやあ、人に自慢するほどじゃありませんが、始めてもう40年になりますね。」

Question

A: 将棋がお強いと伺いましたが。
B: いやあ、人に自慢するほどじゃありませんが、始めてもう40年になりますね。

First, I have problems determining the meaning of 伺う. Does it mean "to ask" or "to be told" here?

Second, I just learned about the comparisons with ほど.
There are two patterns I learned:

八ヶ岳は有名な山だが、富士山ほど有名ではない。

and

田中さんほど仕事がよくできる人はいません。

Now, in context of the dialogue in question, I think that pattern 1 is at play here. Still, considering that I don't really know what the first sentence says, I wonder what to make of this. I also can't find what いやあ means.

Here's my attempt at translation:

A: Have you been told that you're strong at shougi?
B: ? , while I don't boast towards people like that, for the first time already become 40 years.

I also can't make much sense of the second part in the second sentence either

Answer 1

First, I have problems determining the meaning of 伺う. Does it mean "to ask" or "to be told" here?

(～と)伺う here is the humble form (謙譲語) of (～と)聞く, meaning "I heard that~~".
_{～か伺った or ～～かと伺った would be "I asked whether~~."}

Second, I just learned about the comparisons with ほど.
八ヶ岳は有名な山だが、富士山ほど有名ではない。
田中さんほど仕事がよくできる人はいません。

ほど basically means "to the extent (that~)" or "to the (same) extent (as~)". So 富士山ほど有名ではない literally means "not famous to the same extent as Mt. Fuji", hence "is not so famous as Mt. Fuji", and 田中さんほどよくできる人 means "a person who is competent to the same extent as Mr. Tanaka".

And here in your example 人に自慢するほどじゃありません, the ほど also means "to the extent (that~)."

［私の将棋の腕前は］人に自慢するほどじゃありません。
lit.［My shogi technique is］not (good) to the extent that I can boast to others.
I am not good at shogi to the extent that I can boast about it to others.

I also can't find what いやあ means.

It's an interjection (感動詞):

いや〘感〙
驚き・感動・嘆きなどを表す語。いやあ。
「いや、びっくりした。」「いや、すばらしい。」
(明鏡国語辞典)

始めてもう40年になりますね。

始めて here is not "for the first time" (初めて)*, but the te-form of the verb 始める, lit. "started and~" → "since I started". 「～～して+ (period of time) +になる」 means "(period of time) has passed since ~~" or "It's been (period of time) since ~~."

"It's already been 40 years since I started (shogi)."

_{*始めて = [はじめて]{LHHH}, 初めて(for the first time) = [はじめて]{LHLL}}

Answer 2

The sentence means that

A: I heard that you are very good at playing shogi.
B: No, I am not so good that I could boast to anyone, but it's been 40 years since the first time I played.