Q: Will Japanese Apple Store give same care as American Apple Store?

You are likely getting close to or just beyond the 5 year hardware support period from Apple.  I don't mean the warranty which has long expired but the time during which Apple even has parts for a computer such as batteries.

Vintage and obsolete products - https://support.apple.com/HT201624- "Owners of iPad, iPhone, iPod or Mac products may obtain service and parts from Apple or Apple service providers for 5 years after the product is no longer manufactured (or longer where required by law). Apple has discontinued support for certain technologically obsolete and vintage products."

A computer running slowly can be due to various things, most of which are user fixable. Run this analyzer and post the results here.

http://www.etresoft.com/etrecheck

You may be better advised to consider the route to update or replace consumables that

can wear out, before traveling with the five year old MacBook Pro; or at least find estimate

of those costs of quality parts and installation of them. You could perform some yourself.

Examples of sources for replacement and or upgrade parts can include reputable ones

that have served the Apple community for several decades; you need not buy to ask them

questions about what may work. Being able to identify the computer accurately will help.

The model ID number is one of the more common items that facilitates match-up parts.

• How to identify Mac Pro models - Apple Support

upgrade memory, drives, find replacement batteries, etc:

http://eshop.macsales.com/shop/Apple_Upgrade_Guide/Model_ID_MacBookPro_MacBook

However to have a computer evaluated by an Apple Store genius or an authorized Apple

service provider, could give you some ideas on what could be contributing to the issues.

An Apple Store genius may inspect or check the computer; but they don't fix old models.

• Make an appointment at an Apple Retail Store Genius Bar
• Find an Apple Authorized Service Provider
• See all worldwide support telephone numbers

The AASP or authorized service provider can usually locate official parts and restore older

models of Apple products; some can also locate parts from older sources to repair vintage

or obsolete models that are long since seldom ever seen in an Apple Store.

An ifixit.com repair guide for Mac models can be very helpful to take a look into these

devices without opening one of them; helps to see what the level of difficulty may be.

Good luck & happy computing!