一个位老朋友邀请我出席一个宴会。

You can elevate the tone to sound more polite and refined by changing the measure word.

我们很久才没见面了。

I guess you want to emphasize that you haven't met for a long time, so i think it may batter to change"才" into "没". otherwise, 才 seems conflict with 了

他很好客，介绍一下了他新的朋友。

宴会的气氛很热情闹。

"热情" is usually used in describing a person.

虽然我喝得我头疼，但是我认识了一些很好的朋友。