。
On hearing the news, she felt like crying.
Literal
She [topic-は] that news [object-を] hearing [then-て] cry-want feeling [subject-が] did.
泣きたい ('want to cry') is the desiderative ~たい on 泣く. Pairing it with 気がする ('feel that, have a sense that') gives 泣きたい気がする, which translates to English's 'feel like crying.' Note the が rather than を before する — 気 takes が in this idiom, marking the feeling as something that arose rather than something the subject did. The whole construction softens the emotion: she didn't cry, but the impulse was there.