>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~~".  
<sub> ～か伺った or ～～かと伺った would be "I asked whether~~."</sub>

>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 a filler. "Well, ..."

> 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)."

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