kinou nagaku pasokon wo tukatta node, kata ga sukosi itai desu.

With common time words like 昨日 (yesterday), 今日 (today), and 明日 (tomorrow), Japanese often uses them without any particle.

So:

• 昨日 パソコンを使った
• not usually 昨日に パソコンを使った

A particle like に is more common with specific points in time such as 3時に, 月曜日に, or 2024年に.

長い is an -i adjective, and here it needs to modify the verb 使った (used). To do that, you change 長い to its adverb form:

• 長い → 長く

So 長く使った means used for a long time.

This is the same pattern as:

• 早い → 早く
• 遅い → 遅く

Because パソコン is the direct object of 使う.

• パソコンを使う = to use a computer

The thing being used takes を.
So here, the speaker used the computer, making パソコン the object.

Because the sentence describes:

• a past cause: 昨日 長く パソコンを 使った
• a present result: 肩が 少し 痛いです

So the idea is:

• I used the computer for a long time yesterday, so now my shoulder hurts a little.

If the pain were also being described as a past event, you might use 痛かったです instead.

ので introduces a reason or cause, like because or so.

• 使ったので、肩が痛いです
• Because I used it, my shoulder hurts

Compared with から, ので often sounds a little:

• softer
• more explanatory
• less blunt

Very roughly:

• から = because / since
• ので = because / so, with a gentler explanatory tone

Both can work in many situations, but ので is very natural here.

With words like 痛い, the body part usually takes が.

• 頭が痛い = my head hurts
• お腹が痛い = my stomach hurts
• 肩が痛い = my shoulder hurts

That is because 痛い describes a state: the shoulder is painful.
So 肩 is treated as the thing that is in that state, and が marks it.

Japanese often leaves out information that is obvious from context.

Here, if someone says:

• 肩が少し痛いです

it is naturally understood as:

• my shoulder hurts a little

Japanese does this a lot with body parts and personal information.
Instead of saying 私の肩, it often just says 肩, because whose shoulder it is is already clear.

Japanese nouns usually do not mark singular vs. plural unless it matters.

So 肩 can mean:

• shoulder
• shoulders

depending on context.

In natural English, this sentence might be translated as either:

• My shoulder hurts a little
• My shoulders hurt a little

If the speaker wanted to be explicit about both shoulders, they could say 両肩.

少し means a little or slightly. Here it modifies 痛いです.

So:

• 少し痛いです = it hurts a little

It works like an adverb in English.
It makes the statement softer and more precise.

痛い is an -i adjective, and it can already end a sentence by itself:

• 肩が痛い。

Adding です makes it polite:

• 肩が痛いです。

So です here is not changing the basic meaning; it is mainly adding politeness.

The basic meaning would stay the same even if the order changed a bit, because Japanese relies heavily on particles.

For example, this is also natural:

• 昨日 パソコンを 長く 使ったので、肩が 少し 痛いです。

That said, the original order is perfectly fine.
Japanese word order is often flexible, but not completely random. The particles still need to match the roles of the words.