לתעודת הזיהוי שלי יש תוקף עד סוף השנה, אבל הכרטיס הישן כבר לא תקף.

This is a very common Hebrew pattern:

ל־... יש ... = ... has ... / there is ... for ...

So:

• לתעודת הזיהוי שלי = for my ID card
• יש תוקף = there is validity

Literally, the clause is something like:

For my ID card, there is validity until the end of the year.

Natural English: My ID card is valid until the end of the year.

Hebrew often uses יש plus ל־ instead of a direct verb like to have.

יש תוקף literally means has validity or there is validity.

• יש = there is / there are
• תוקף = validity, legal force, effect

So לתעודת הזיהוי שלי יש תוקף means:

• literally: My ID card has validity
• naturally: My ID card is valid

This is a slightly more formal way to talk about validity. In everyday speech, people may also simply say something like:

תעודת הזיהוי שלי תקפה עד סוף השנה
= My ID card is valid until the end of the year

They are related, but they are different parts of speech.

• תוקף = a noun: validity, force
• תקף = an adjective: valid, in force

So:

• יש תוקף = has validity
• לא תקף = not valid

That is why the sentence uses both:

1. לתעודת הזיהוי שלי יש תוקף...
   = My ID card has validity...

2. ...אבל הכרטיס הישן כבר לא תקף.
   = ...but the old card is no longer valid.

Because this is a construct phrase (called smikhut in Hebrew), where two nouns are linked together:

• תעודת = document/card of
• הזיהוי = the identification

So תעודת הזיהוי means the identification card/document.

In Hebrew, when two nouns form this kind of phrase:

• the first noun changes into the construct form
• the definiteness usually appears on the second noun

That is why you get:

• תעודת זיהוי = an ID document / identification card
• תעודת הזיהוי = the identification card

Not תעודה הזיהוי.

Hebrew possessive forms with של usually come after the noun:

• הספר שלי = my book
• תעודת הזיהוי שלי = my ID card

So Hebrew does not normally say the equivalent of my ID card with my before the noun the way English does. Instead, it says:

the ID card of me → תעודת הזיהוי שלי

This is the normal modern Hebrew pattern.

Because סוף השנה is another common construct phrase:

• סוף = end
• השנה = the year
• סוף השנה = the end of the year

So:

עד סוף השנה = until the end of the year

This is the most natural Hebrew wording.
A phrasing like עד הסוף של השנה is much less natural here and sounds heavier.

Hebrew often prefers these compact construct phrases:

• תחילת היום = the beginning of the day
• אמצע השבוע = the middle of the week
• סוף השנה = the end of the year

Because in the present tense, Hebrew usually does not use a separate verb meaning is/are.

So:

• הכרטיס הישן תקף = the old card is valid
• הכרטיס הישן לא תקף = the old card is not valid

Hebrew simply puts the subject and the adjective together, without a present-tense to be.

This is one of the most important patterns in Hebrew grammar.

Because תקף agrees with הכרטיס, and כרטיס is masculine singular.

• כרטיס = masculine
• therefore: כרטיס תקף = a valid card

If the noun were feminine, you would use תקפה:

• תעודה תקפה = a valid certificate/document

So in this sentence:

• הכרטיס הישן ... תקף is correct because כרטיס is masculine.

Because in Hebrew, when a noun is definite, the adjective describing it is also usually definite.

So:

• כרטיס ישן = an old card
• הכרטיס הישן = the old card

Both words take definiteness:

• הכרטיס = the card
• הישן = the old

This is a standard rule in Hebrew adjective agreement.

כבר לא means no longer.

So:

• לא תקף = not valid
• כבר לא תקף = no longer valid

The word כבר often means already, but together with לא it commonly means not anymore / no longer.

Examples:

• אני כבר לא גר שם = I no longer live there
• זה כבר לא חשוב = That is no longer important

So in your sentence, כבר לא תקף means the old card used to be valid, but now it is not.

אבל means but.

It connects the two contrasting ideas:

• My ID card is valid until the end of the year
• but the old card is no longer valid

So it marks a contrast between:

1. the current document still being valid
2. the old card not being valid anymore

Yes. A very natural alternative would be:

תעודת הזיהוי שלי תקפה עד סוף השנה.

That means the same thing:
My ID card is valid until the end of the year.

The difference is mainly style:

• יש תוקף = a bit more formal, slightly more official-sounding
• תקפה = more direct and simple

Both are correct.

Note that with תעודת הזיהוי, you would use תקפה because תעודה is feminine.

It is understandable, but in everyday Israeli Hebrew, the most common term for the Israeli identity card is usually:

תעודת זהות

That said:

• תעודת זיהוי means identification document / ID card
• תעודת זהות is the especially common everyday term for an official identity card

So a learner might hear both, but תעודת זהות is often the more familiar everyday expression.