Nagoya Airport to Takayama

I will be arriving Nagoya Airport around 8am. And would like to head straight to Takayama. What is the best way from the airport to Takayama? We are travelling with 2 children. And how long will it take to arrive Takayama?

Take the Meitetsu airport train to Nagoya and then the JR limited express train to Takayama. The one way trip takes about 3.5 hours and costs around 7000 yen.

More options:
http://www.japan-guide.com/e/e5908.html?aFROM=2155_5900

You must mean Central Japan International Airport [NGO], nicknamed "Centrair."

If you hope to get a Japan Rail Pass, then exit Meitetsu at Meitetsu Nagoya Station and walk to Nagoya Station of JR-Central.
If you hope to go with regular tickets, then it sounds better to make a Meitetsu - JR transfer at Kanayama Stations (adjoining in the same building).

The time for getting through immigration procedures (and customs) is not predictable;
if your flight is not delayed and everything is done smoothly,
then you will catch a Ltd. Express train to reach Takayama Station at 13:07 or at 14:16.

Central Japan International Airport Sta. of Meitetsu
| 32 min on a Ltd. Express, 43 min on a Semi Express;
Kanayama Sta. of Meitetsu
| about 25 min: for exiting Meitetsu through one of Meitetsu Line Ticket Gates,
| purchasing JR tickets at a ticket office near JR Ticket Gate,
| entering JR through the gate and catching a train;
| http://english.jr-central.co.jp/info/station/kanayama.html
Kanayama Sta. of JR-Central: Track 2 (shown in green in this map)
| within 5 min on Chuo Line;
Nagoya Sta.: about 10 to 55 min for changing trains
| about two and a half hours on a Ltd. Express;
Takayama Sta.

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It does not take 25 minutes exiting Kanayama Meitetsu and changing to JR. Try 5 minutes.

The link Uji provides is a good guide Nuzst.

Train schedule and price finder:
http://world.jorudan.co.jp/norikae/cgi-bin/engkeyin.cgi
From "Central Japan Intl Airport"
To "Takayama"

You can try reserving Meitetsu seats in a special express called "MU SKY" (fully with reserved seat cars) or a Ltd. Express (partly with reserved seat cars).
http://www.meitetsu.co.jp/english/surcharge/index.html
(Personally I think little children can get bored while waiting twice within one hour or so for their parent's reserving seats and purchasing tickets.)

: It does not take 25 minutes exiting Kanayama Meitetsu and changing to JR. Try 5 minutes.

"5 minutes" may be enough for just moving gate to gate; "about 25 min" is an estimation including time for purchasing long-distance JR tickets at a person-to-person counter in the JR Ticket Office.
Let me repeat, to avoid misunderstanding.
about 25 min: for exiting Meitetsu, purchasing JR tickets, entering JR through and catching a train.

: The link Uji provides is a good guide

I agree.
A highway bus (mentioned in the page) may be another good option,
though Meitetsu Bus Center (for booking and catching a bus) seems to be a bit far from Meitetsu Nagoya Station.

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Don't get off at Kanayama that doesn't make any sense. The next stop after when coming from the airport is Nagoya. Change and get your ticket in Nagoya to save time.

: Don't get off at Kanayama that doesn't make any sense.

Personally I see no reason for any of the two routes to be definitely discarded.
That's not our choice.
The way one person thinks as best / worst is not necessary the best / worst way for another.

: The next stop after when coming from the airport is Nagoya. Change and get your ticket in Nagoya to save time.

The next stop is Meitetsu Nagoya Station, not Nagoya Station where long-distance JR tickets are available. They are located near enough for one who knows the way, but they are not adjoining in the same building.
Waking pace differs by persons and situations. The time for purchasing tickets is not predictable. No one can foretell which of the two routes will be more time-saving as a result.

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Personally I see no reason for any of the two routes to be definitely discarded.

I can see one reason, clearly you have not been to Nagoya before. There is absolutely no reason to get off at Kanayama. There is no positive about it at all its just going to add extra time to his/her journey.

The next stop is Meitetsu Nagoya Station, not Nagoya Station where long-distance JR tickets are available. They are located near enough for one who knows the way, but they are not adjoining in the same building.

Again, you clearly have not been there. The two are very accessible and very clearly marked not to mention well connected.
When travelling with kids you want to change trains the least amount of time and not complicate the journey.

No need to complicate this persons Journey.

Nuzst look at the route finder that K linked, that is the most efficiant way for you to take.
I'm from Nagoya :)

If Guest kindly indicates how to get from Meitetsu Nagoya Sta. entrance to Nagoya Sta. JR Ticket Office, I think it will help.

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Thanks to an abundance of signs, it is absolutely no problem to go from Meitetsu Nagoya Station to JR Nagoya Station and find a ticket counter there. There could be some stairs without escalators, though.

If Guest kindly indicates how to get from Meitetsu Nagoya Sta. entrance to Nagoya Sta. JR Ticket Office, I think it will help.

5 Minute walk following the clearly marked English signs on the way to the Large well manned JR ticket office. A plotted course on a map is not required, Nagoya really is well marked.