Person A : "I am sick today"
Person B : "Oh no!"
How do the Japanese express feelings of negative shock towards bad news? What would Person B say in Japanese?
Depends on the case - there are many ways to express a negative surprise.
Example using 大変ですね which expresses some, but not excessive sympathy and cannot be used for "really bad" news. In some case can sound ironic, so be careful:
- 今朝、寝坊しました。// I overslept today
- 大変ですね！ - // literally "so hard isn't it?", actually kind of "i am sorry for you" but without excessive sympathy.
Another example with 大変, this time much stronger one:
- 火事だ！ // It's fire!
- あら！大変！ // literally "hard", actually kind of "oh no!"
An example using 残念:
- ハズレだ！ // (the lottery ticket) did not win
- 残念ですね。// literally "regretful, isn't it?", actually "I am sorry for you".
An example using どうしよう:
- 津波だ！逃げろ！ // Tsunami is coming! Run!
- やばい、どうしよう。 // literally "horrible, what should I (we) do?", actually expresses some distraction and (negative) surprise.
An example using びっくり:
In this situation a person is surprised by a something in the dark but then it turned out it was a cat: - びっくりした！！！ // Literally means "I was surprised!", actually it means "I was surprised" but expressing kind of relief.
As for actual "I am sick today" the following would sound very natural:
- 風邪をひいてしまった。 // I'v got a cold today
- 大丈夫？お大事に！ // Are you ok? Get well soon! (お大事に) is an etiquette word used to express sympathy for one's sickness. The full variant is お大事にしてください。
An example using 嫌だ (いやだ), heard extremely often, especially among women:
- 施術しなければなりません。// You must do surgery...
- 嫌だ〜！ // Literally "I hate it", actually a common expression of negative surprise.
An example using 嘘 (うそ), used quite often too:
- ○○ちゃんは事故った。 // Ms. ○○ got into an accident.
- 嘘？ // Literally "lying?", actually a common expression of negative surprise and disbelief.
Finally, an examples which is little rude, but still heard very often:
- やっぱり、動かないんだ。まいった! // Apparently it does not work. What a bummer!
That depends on the actual thing they’re expressing shock toward, formality, etc. just like in any language.
Various interjections include まさか, しまった (my go-to interjection when something upsets me and I’d say “oh balls” or “aw shit” in English), うわぁ, うそだろう, あら（ま）, 大変だ, etc.
Something like “I’m sorry for your loss” as a reaction to very bad news would be お悔やみを申し上げます.
As for some sort of unfortunate event, there's the good old 気の毒 as well. 可哀そう could work in certain circumstances too. Or, as mentioned, 残念 too. It's all very circumstantial as user1713450 points out.
Sigh... my cat kicked the bucket yesterday.
What! You're shitting me? He was so full of life the last time I saw him... that's terrible.
Ah man, I'm dying. I feel like I caught a cold.
Poor you. Don't ya think maybe you've been pushing yourself a bit too hard recently? Hurry up and get home and get some rest.
Why not in standard Japanese? Because it's more fun and "real" this way.