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 a filler. "Well,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}