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> 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
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question
grammar meaning
「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。」
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.
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answer
> 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}
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