Minun ranteeni parantuu hitaasti, mutta se on jo parempi kuin eilen.

It can feel redundant to an English speaker, but in Finnish this is normal.

• minun = my (genitive form of minä, I)
• -ni = the possessive suffix meaning my
• ranteeni = my wrist

So minun ranteeni literally has both a possessive pronoun and a possessive suffix.

In Finnish, you can often say either:

• ranteeni = my wrist
• minun ranteeni = my wrist (more explicit, often a bit clearer or more emphatic)

In standard Finnish, using both is very common and completely correct.

Because the noun ranne changes its stem before many endings are added.

The basic form is:

• ranne = wrist

But when you add endings, the stem becomes rantee- / rante-:

• ranteen = of the wrist
• ranteessa = in the wrist
• ranteeni = my wrist

So this is just part of how this noun type behaves. You are not simply attaching -ni directly to the dictionary form.

Parantuu means gets better, heals, or recovers.

It comes from the verb parantua, which is an intransitive verb. That means the subject improves by itself:

• Ranne parantuu. = The wrist is healing / getting better.

Compare that with parantaa, which is transitive and means to improve or to heal something:

• Lääkäri parantaa potilaan. = The doctor heals the patient.

So here parantuu is the correct choice because the wrist itself is healing.

Finnish often uses the simple present where English uses either the simple present or the present continuous.

So:

• Ranteeni parantuu hitaasti
  can mean
• My wrist heals slowly
• My wrist is healing slowly

In this sentence, the natural English translation is is healing slowly, because it describes an ongoing process.

Because hitaasti is an adverb, while hidas is an adjective.

• hidas = slow
• hitaasti = slowly

The verb parantuu needs an adverb here, because we are describing how the healing happens:

• Ranne parantuu hitaasti. = The wrist heals slowly.

A very common Finnish pattern is:

• adjective + -sti → adverb

For example:

• nopea → nopeasti = quick / quickly
• hidas → hitaasti = slow / slowly

Se means it here, and it refers back to ranteeni (my wrist).

So:

• mutta = but
• se = it
• on = is

Together:

• mutta se on jo parempi = but it is already better

Finnish often uses se for things, animals, and even people in everyday speech, but here it is simply the normal word for it.

Jo means already.

So:

• se on parempi = it is better
• se on jo parempi = it is already better

The word jo adds the idea that there has been progress by now, even if the healing is still slow.

So the full sentence has a nice contrast:

• healing is slow
• but there is already some improvement

Parempi is the comparative form of hyvä (good).

• hyvä = good
• parempi = better
• paras = best

So:

• se on parempi = it is better

This is an irregular comparison, just like English good → better → best.

Because parempi is an adjective, and here it describes the noun wrist.

• parempi = better (adjective)
• paremmin = better (adverb, as in more well)

Compare:

• Ranne on parempi. = The wrist is better.
  (describes the wrist)

• Ranne paranee paremmin nyt. = The wrist is healing better now.
  (describes how it heals)

In your sentence, the meaning is it is better, so parempi is correct.

Not literally. This is a very common kind of shortened comparison.

• kuin = than
• eilen = yesterday

So parempi kuin eilen literally looks like better than yesterday, but the real meaning is:

• better than it was yesterday

Finnish, like English, often leaves out the repeated part because it is understood from context.

So this is perfectly natural and idiomatic.

Because eilen is already an adverb meaning yesterday.

It is not behaving like a regular noun here, so it does not need a case ending.

Compare:

• eilen = yesterday
• tänään = today
• huomenna = tomorrow

These are time adverbs and are often used just as they are:

• Se sattui eilen. = It happened yesterday.
• Se on parempi kuin eilen. = It is better than yesterday / than it was yesterday.

The given word order is very natural:

• Minun ranteeni parantuu hitaasti, mutta se on jo parempi kuin eilen.

Finnish word order is more flexible than English, but changing it usually changes emphasis.

For example:

• Ranteeni parantuu hitaasti, mutta se on jo parempi kuin eilen.
  Also very natural.

• Hitaasti minun ranteeni parantuu...
  Possible, but much more marked and less neutral.

So the original sentence uses a normal, neutral word order.

Yes. That would still be correct and natural.

• Ranteeni parantuu hitaasti... = My wrist is healing slowly...

In fact, Finnish often leaves out the possessive pronoun when the possessive suffix already shows the meaning.

So both are good:

• Minun ranteeni parantuu hitaasti...
• Ranteeni parantuu hitaasti...

The version with minun is a bit more explicit.