Some general compliments:
"Мне нравится твоя прическа / Mne nravitsya tvoya pricheska" is "I like your hairstyle"
"У тебя красивые глаза / U tebya krasivye glaza" is "Your eyes are pretty"
Cool Russian phrases... I'm not sure what to tell you here, like, what would you want to say? What do you even think is cool?
Haha, sorry to make you overthink it. I just wondered what might sound cool and funny, like a pun or funny word because we sit next to each other every day. Don't worry too much
Ёлки-палки or Yolki palki - for expressing surprise, like Holy cow. I would do a double-take if any non-Russian speaker says it to me.
You can also try asking her what Бычить / Bycheet means. That would be an interesting conversation if she doesn't know English.
A pun:
Капитализм - Вашингтон, Коммунизм - Нашингтон / Kapitalizm - Vashington, Kommunizm - Nashington
The meaning of the joke: Capitalism - New York, Communism - New Ourk