First, the answer to your question is "It's impossible even for Japanese people (to correctly read all of them)."
The reading for the example you posted would be "Matsutouya Yumi."
It's 松 (Matsu) 任 (tou) 谷 (ya) 由 (Yu) 美 (mi).
The reading for 任 (tou) is a bit hard, but you can take and read each character by itself.
There are also names that get special readings when there are two or more characters combined.
服部(Hattori), 飛鳥(Asuka), 小鳥遊(Takanashi)
服部 isn't 服(Hat)部(tori); it's only read Hattori when it's written 服部.
The same characters can also have different readings.
中田(Nakata, Nakada), 宏紀(Hironori, Hiroki, etc), 篤(Atsushi, Atsumu, etc)
There are also names which are pronounced the same but which use different characters.
Nakata (中田, 仲田, 名方), Hironori (宏紀, 裕典, 博憲, etc), Atsushi (篤, 敦, 厚志, etc)
According to Japanese law, it's possible to register a person's name with whatever reading you want.
So there will be times when weirdo parents register a reading that's totally unrelated to the characters they used for their child's name.
I'll propose three ways to read names.
1) When you can copy and paste a name, do a Google search with the name. (That's serizawa's method.)
2) If you can enter in even one kanji, use the site given below.
http://kanji.reader.bz/
http://kanji.reader.bz/任
http://kanji.reader.bz/more/任
3) If you can't enter in even one character, ask a Japanese person.
When you read kanji in Japanese, there's something called "On'yomi" and something called "Kun'yomi."
I'll explain by using 山 (san, yama) as an example.
Kanji were transmitted over to Japan from China.
In short, the shape (山), meaning (mountain), and Chinese pronunciation (san {shan}), were all brought over at the same time as a set.
This is "On'yomi."
"On'yomi" means it's the reading that's similar to the Chinese pronunciation.
山 (san).
However, before the character for "mountain" was brought to Japan, Japanese called mountains "yama."
Then the character meaning "mountain" 山 was introduced.
Therefore, 山 isn't pronounced just "san," but also "yama."
This is "Kun'yomi."
"Kun'yomi" means it's the reading that signifies the original Japanese meaning.
山 (yama.)