Я написал адрес по памяти, но потом проверил его.

Both verbs are past tense masculine singular forms.

• написал = wrote
• проверил = checked / verified

In Russian past tense, the form agrees with the subject in gender and number:

• написал = masculine
• написала = feminine
• написало = neuter
• написали = plural

Since the speaker is understood as masculine here, the sentence uses написал and проверил. If the speaker were female, it would be:

Я написала адрес по памяти, но потом проверила его.

They are perfective because the sentence talks about completed actions:

• first, the person wrote down the address
• then, the person checked it

Perfective verbs present the actions as whole, finished events.

Compare:

• Я писал адрес... = I was writing the address / I wrote the address (with focus on process, repetition, or background)
• Я написал адрес... = I wrote down the address (completed result)

And similarly:

• проверял = was checking / used to check
• проверил = checked, finished checking

This sentence is basically a sequence of completed steps, so perfective is the natural choice.

По памяти means from memory or by memory, in the sense of relying on what you remember rather than looking something up.

So:

• написать адрес по памяти = to write the address from memory

This expression is very common in Russian. You can also use it with other actions:

• рассказать по памяти = to recite from memory
• нарисовать по памяти = to draw from memory
• восстановить по памяти = to reconstruct from memory

The preposition по often has many meanings in Russian, and this is one fixed, natural use you should learn as a phrase: по памяти.

Because адрес is the direct object of написал, and for an inanimate masculine noun in the singular, the accusative is the same as the nominative.

So:

• nominative: адрес
• accusative: адрес

That is why the form does not change.

Compare with a feminine noun:

• Я написал записку. Here записка becomes записку in the accusative.

And compare with an animate masculine noun:

• Я видел брата. Here animate masculine accusative changes and looks like the genitive.

So адрес stays адрес because it is masculine, singular, inanimate.

Его means it here, referring back to адрес.

In Russian, object pronouns reflect grammatical gender:

• masculine: его = him / it
• feminine: её
• neuter: его
• plural: их

Since адрес is masculine, the sentence says проверил его = checked it.

Note that его can mean either him or it, depending on context. Here it clearly means it, because it refers to адрес.

Russian often omits a repeated subject when it is obvious from context.

So:

• Я написал адрес по памяти, но потом проверил его.

naturally means:

• I wrote the address from memory, but then I checked it.

You could repeat я:

• Я написал адрес по памяти, но потом я проверил его.

But that usually sounds more emphatic, contrastive, or stylistically heavier. In ordinary speech and writing, Russian often leaves the second я out because it is already understood.

Потом means then / afterward / later.

It marks the sequence clearly:

1. написал адрес по памяти
2. потом проверил его

Without потом, the sentence would still make sense:

• Я написал адрес по памяти, но проверил его.

But that sounds less natural and less clear in the intended sequence. Потом emphasizes that the checking happened afterward, not at the same time.

Because но means but, and in Russian it normally joins two clauses with a comma before it.

So:

• Я написал адрес по памяти, но потом проверил его.

This is standard punctuation, just like English often uses a comma before but when joining two full clauses.

Yes, but the meaning or tone would change.

• но = but
  This shows a mild contrast: the speaker wrote it from memory, but then checked it anyway.

• а can also express contrast, but often a softer one, more like whereas / and yet / while
• и = and, which would simply connect the actions without highlighting contrast

Examples:

• Я написал адрес по памяти, но потом проверил его.
  I wrote the address from memory, but then checked it.

• Я написал адрес по памяти, а потом проверил его.
  I wrote the address from memory, and then / whereas then I checked it.
  This sounds very natural too, but it feels less adversative than но.

• Я написал адрес по памяти и потом проверил его.
  I wrote the address from memory and then checked it.
  This focuses more on simple sequence.

So но suggests: I relied on memory, but I still decided to verify it afterward.

Not exactly, though they can be close in meaning.

• написал адрес = wrote the address
• записал адрес = wrote down the address / noted the address down

In many everyday situations, записать адрес is very common when you mean writing information down for practical use.
Написать адрес is also possible, but can sound a little more neutral or literal: you wrote the address.

So the original sentence is correct, but in some contexts a native speaker might also say:

• Я записал адрес по памяти, но потом проверил его.

That would slightly emphasize the act of noting it down.

Yes, Russian word order is fairly flexible, but different orders change emphasis.

The neutral version is:

• Я написал адрес по памяти, но потом проверил его.

Other possible orders include:

• Адрес я написал по памяти, но потом проверил его.
  Emphasizes address

• Я по памяти написал адрес, но потом проверил его.
  Emphasizes from memory

• Но потом я его проверил.
  Emphasizes what happened afterward

Even though Russian allows this flexibility, not every order is equally natural in every context. The original sentence is a very normal, neutral way to say it.

It can move, because Russian word order is flexible, but the original position is very natural:

• Я написал адрес по памяти...

You could also say:

• Я по памяти написал адрес...
• Адрес я написал по памяти...

All of these are grammatical. The difference is mainly emphasis:

• по памяти after адрес is neutral
• по памяти earlier in the sentence gives it more focus

So it does not have to stay in exactly that place, but the original order is a good default.

Russian has no articles like a, an, or the.

So адрес can mean:

• an address
• the address

The exact meaning is understood from context. In this sentence, English usually needs the address because it is a specific address already known in the situation, but Russian simply says адрес.

This is normal for all Russian nouns:

• Я купил книгу. = I bought a/the book.
• Он открыл дверь. = He opened the door.

Context does the work that articles do in English.