Я хочу примерить простую рубашку без яркого узора.

This is a question about aspect.

• примерить = perfective
• примерять = imperfective

In this sentence, Я хочу примерить..., the speaker wants to do one complete try-on of the shirt. That makes примерить very natural.

Compare:

• Я хочу примерить эту рубашку. = I want to try on this shirt.
• Я люблю примерять одежду. = I like trying on clothes.
• Она долго примеряла платье. = She was trying on the dress for a while.

So примерить here focuses on a single, concrete action.

Because it is the direct object of the verb примерить, so it goes into the accusative case.

Base form:

• рубашка = shirt

Accusative singular:

• рубашку

So:

• Я хочу примерить рубашку. = I want to try on a shirt.

This is a very common pattern in Russian:

• читать книгу
• купить машину
• надеть куртку

All of those objects are in the accusative.

Because adjectives in Russian must agree with the noun they describe in:

• gender
• number
• case

Here, рубашку is:

• feminine
• singular
• accusative

So the adjective must match:

• простая рубашка = plain/simple shirt (nominative)
• простую рубашку = plain/simple shirt (accusative)

That is why both words change together.

Простой can mean different things depending on context.

Common meanings include:

• simple
• plain
• not complicated
• basic

With clothing, простая рубашка usually means:

• a plain shirt
• a simple shirt
• a basic shirt
• not fancy / not elaborate

So here it does not mean the shirt is easy. It means the shirt is simple in style.

Because the preposition без always takes the genitive case.

So:

• узор → узора
• яркий → яркого

That gives:

• без яркого узора = without a bright pattern

This is a very important rule:

• без сахара = without sugar
• без воды = without water
• без проблем = without problems

After без, the following word or phrase must be in the genitive.

Because яркого is the genitive singular masculine/neuter form of яркий, and it agrees with узора.

Here is the pattern:

• nominative: яркий узор
• genitive: яркого узора

Since узор is masculine singular and is required by без to be in the genitive, the adjective changes too.

Both can be correct, but they feel slightly different.

• без яркого узора = without a bright pattern
  This sounds like without any single bright, noticeable pattern/design
• без ярких узоров = without bright patterns
  This sounds a bit more general or plural

In many shopping situations, both would be understood well.

The singular version in your sentence is very natural if the speaker means:

• no bold design
• no loud pattern
• nothing visually flashy

Because Russian often expresses that idea inside the verb itself.

• примерить already means to try on

So Russian does not need a separate word matching English on here.

Compare:

• примерить рубашку = try on a shirt
• примерить обувь = try on shoes
• примерить пальто = try on a coat

English uses a verb + particle (try on), but Russian uses one verb.

This is the standard pattern:

• хочу + infinitive = want to do something

So:

• Я хочу примерить... = I want to try on...

More examples:

• Я хочу купить это. = I want to buy this.
• Я хочу посмотреть. = I want to look.
• Я хочу уйти. = I want to leave.

So grammatically, примерить is the infinitive that depends on хочу.

No, Russian word order is fairly flexible, but the version you have is neutral and natural:

• Я хочу примерить простую рубашку без яркого узора.

That is probably the best default order.

You can move words around for emphasis, but some versions sound more marked than others. For example:

• Я хочу примерить рубашку без яркого узора.
• Простую рубашку без яркого узора я хочу примерить.

The second one is possible, but it sounds more emphatic or stylistically marked. For learners, the original order is the safest choice.

Not always. Рубашка usually means a shirt, often more like a button-up or collared shirt.

Other common clothing words are:

• футболка = T-shirt
• блузка = blouse
• майка = tank top / undershirt
• кофта = sweater/top/jacket-like casual upper garment, depending on context

So рубашка is correct if the person means a shirt rather than a T-shirt or blouse.

A natural stress pattern is:

Я хочу́ приме́рить просту́ю руба́шку без я́ркого узо́ра.

Word by word:

• хочу́
• приме́рить
• просту́ю
• руба́шку
• я́ркого
• узо́ра

Stress is important in Russian, so it is useful to learn words together with their stress.