The Lonely Planet has changed a lot since 1998 – authors, topics, etc. While there can be seemingly endless (at least they feel endless) discussion comparing the older version to the newer one, the fact is that stores, restaurants, hotels, and other places change, so a new book is not a bad investment.
The MRT system is not comprehensive, but it has expanded quite a bit in the intervening years. If the issue is protecting your child from UV exposure (as opposed to diseases like EPP that involve light exposure in the visible spectrum), there will be other publicly available transport means to get from point A to Point B that will be aboveground and have windows but which will not lead to tropical-style sunburns. Taxis are also quite a bit cheaper in Taiwan than in many other parts of the world, so that can also be a good option. If the problem is more of an EPP-type nature, then there are significant parts of the MRT system going east and west that remain completely below ground – some parts of Taipei also have “underground shopping streets” that may also allow some walking (normally parallel to an MRT line) without sun exposure.
For city-to-city travel, the relatively new High Speed Rail has been a godsend, although there are significant and beautiful parts of the country that the system does not reach. I recall these have shades that can be drawn to protect your child.
The Taipei 101 building’s observation deck’s audioguide device very nicely has a different nighttime explanation from daytime, which would allow you to visit, listen to the audioguide and then point things out to your child. Eveningtime explorations may also work in three other ways: 1) cooler at night (watch out for mosquitos, though); 2) lots of good food at night markets and things to see/buy as well; and 3) perhaps you will have less adaptation to the time zone.