Good work! It's hard to hear a lot of parts in this song; there are some funky rhythms that disguise a lot of syllables. After listening to it a couple times, here are the lyrics as I hear them, and their translation.

rainy day

忘れたくない事なら覚え様としなくても
忘れる事なんてないそう静かに感じる

あれはある寒い日で強い雨の中
僕はただ君だけを待ち続けていた

もし今が僕の終わりだったとしたら　それでも
構わないと思える程何も怖くなかった

幸せに笑い合う人々が　僕の
すぐ側を何度でも通り抜けたけど

ひとりきり傘もなく立ち尽くす僕は
誰よりも穏やかに微笑んで行った

凍える手をかざして
現れた君を見て
泣き出しそうになったのは
悲しみの所為じゃない

僕達を？？？？？光だけ増した
これからの行く道を祝うかの様に

--------

Wasuretaku nai koto nara oboe you to shinakute mo
Wasureru koto nante nai sou shizuka ni kanjiru

Are wa aru samui hi de tsuyoi ame no naka
Boku wa tada kimi dake wo machi tsudzukete ita

Moshi ima ga boku no owari datta to shitara Sore de mo
Kamawanai to omoeru hodo nanimo kowaku nakatta

Shiawase ni warai au hitobito ga Boku no
Sugu soba wo nando de mo toori nuketa kedo

Hitori kiri kasa mo naku tachi tsukusu boku wa
Dare yori mo odayaka ni hohoende itta

Kogoeru te wo kazashite
Arawareta kimi wo mite
Naki dashi sou ni natta no wa
Kanashimi no sei ja nai

Boku tachi wo ????? hikari dake mashita
Kore kara no yuku michi wo iwau ka no you ni

--------

Even if I don't try to remember what I don't want to forget,
I peacefully feel that I never really forget it.

That was one cold day; I just kept on waiting
In the strong rain, just for you.

If right now happened to be the end of me,
I wouldn't be afraid at all, to the point that it would seem I didn't care.

People that happily smiled at me
Blew right by me so many times.

But, standing still by myself, without even an umbrella,
I went on smiling more peacefully than anyone.

When I shaded my eyes with my freezing hand *
To see you appear,
It wasn't sadness
That made me feel like crying.

Only the light that ____ us grew,
As if celebrating the road we'd take from then on.

* "te wo kazasu" means to hold your hand, flat, across the top of your eyes as you strain to look over a distance, usually to block offensive light out of your line of vision. I didn't really know how we say that in simple English...

I can't make out the last line. I'm hearing "igi shiteru", but that doesn't really make sense ["Only the light that objects to us grew"]. Anyone else know what it might be?

This song is so simple, but that makes it so sweet, I think. It's just Ayu waiting out in the rain for someone, but being more than happy to wait for this person. I like it very much.