You may not know thise plant ume but its fruit is really popular in Japan.

'This' is fine, but 'the' sounds a bit better here.

Umeboshi, or literally, “dried ume” is a really popular pickle in Japan, and many Japanese people have it for breakfast, although its sour taste and strange softness make it difficult to recommend it to you.

Ume is also known for its blossom’s' beauty.

「Blossoms'」in the plural sounds better here.

So popular that ume blossoms were even more popular than sakura blossoms (cherry blossoms) in Manyoshu, which iwas composed in 759 and is the oldest compilation of Japanese poetry.

Use past tense for when it was composed. It's fine to use present tense to say 'it is the oldest compilation of Japanese poetry', though, because it still is.

However, sakura blossoms became more popular than ume blossoms in the middle of hHeian period (794 to 1185) and eventually sakura became Japan’s symbol.

Why am I talking about ume, you ask?

UThe ume plums were already ripe, so my father asked me to pick them with him before they faell off the tree.

Initially, the starting time was supposed to be 10 o’clock, but I overslept so I started picking them at 10:45.¶
OR: Initially, we were meant to start at 10 o'clock, but I overslept so I started picking them at 10.45.

This is fine, but I've also given an option that would sound more natural.

The weather was so pretty that I took several pictures of the tree and, the ume and the sky.

It’s a shame that I cannot upload them becaussince I’m not a premium user here.

'Because' is fine, but 'since' sounds a little more natural to me.

I didn’t often climb trees as a child, and as an adult I haven’t done for several years, so I was really scaryed.

'Scary' means you yourself frighten other people. 'Scared' means you feel like something else is frightening.
For example, 'Climbing trees is scary' and 'I am scared of climbing trees'.

Making matters worse, I recently seldom go outside recently, so my strength gothas weakened.

'Got weakened' just sounds a bit unnatural.

I made sure that I gripped the stable branches tightly so that I don’idn't fall off the tree.

Two hours had passed and we had picked almost all the ume plums onfrom the tree.

In a big jar, wWe put ume plums, vinegar, sugar cubes, and salt in a big jar, and wait for the drink isto be done.

This sounds more natural.

'Is done' is ungrammatical.

It’s not alcoholic and sweet, but we can make it an alcoholic drink if we add liquor.

「酒じゃなくて甘くない」という意味を書くつもりなら、'It's not alcoholic or sweet' のほうがいいです。
「酒じゃなくて甘い」という意味を書くつもりなら、'It's sweet, but not alcoholic' のほうがいいです。

We had too much ofany ume plums, so my grandmother, who was near me and my father during thwhile we were picking, gave a kilogram of ume plums to our neighbors and her friend who came.

'Many' not 'much' here.

'While we were picking' sounds more natural than 'during the picking'.

Ume pPlum pPicking

Nouns and verbs are capitalized in titles.

You may not know thishave heard of the ume plant ume but its fruit is really popular in Japan.

Umeboshi, or literally, “dried ume” is a really popular pickle in Japan and many Japanese people have it for breakfast, though the sour taste and strange softness of it make it difficult to recommend it to you.

So popular that ume blossoms were even more popular than sakura blossoms (cherry blossoms) in the Manyoshu, which iwas composed in 759 and is the oldest compilation of Japanese poetry.

You could also write 759 A.D. or 759 C.E. to make it more clear what year you're talking about.

However, sakura blossoms became more popular than ume blossoms in the middle of hthe Heian period (794 to 1185) and eventually sakura became Japan’s symbol.

Heian is a proper noun and so is capitalized.

UThe ume plums were already ripe, so my father asked me to pick them with him before they fall off the tree.

Initially, the starting time was was originally supposed to bestart at 10 o’clock, but I overslept, so I started picking them at 10:45.

The weather was so pretty that I took several pictures of the tree and, ume, and sky.

We had to climb the tree because ume plums are attached to the higher branches.

I didn’t often climb trees as a child, and asnow that I'm an adult I haven’t done for several years, so I was really scaryed.

Should be either "it was really scary" or "I was really scared."

Making matters worse, I recently seldom go outside lately so my strength got weakehas waned.

I made sure that Io gripped the stable branches tightly so that I don’wouldn't fall ofut of the tree.

Two hours had passed and we hadWe picked almost all the ume plumes on the tree in two hours.

We gatherfilled three cardboard box full ofes with ume plums.

So what do we do with them?

In a big jar, wWe put ume plums, vinegar, sugar cubes, and salt andin a big jar and then wait fortill the drink is done.

It’s not alcoholic and sweetsweet and non-alcoholic, but we can make an alcohol drink if we add liquor.

We had too much ofany ume plums, so my grandmother, who was near me and my father during the picking, gave a kilogram of ume plums to our neighbors and her friend who also came.