There is a lot of way to mark a point of interest, you can almost litterally use guess what with « vous ne devinerez jamais », or for a really casual locution to one person « devine quoi, … ». You can also use « saviez vous que ».

As for « tenez-vous que … » and « figurez-vous bien » are not correct expression, « tenez-vous bien » can be translated as “hold yourself tight” and « tenez-vous que … » would have no meaning for French people.

However, yes you can use « tenez-vous bien » or « tiens-toi bien » right before a point of interest like you do in your example, but i am not quite sure what is your question.

I hope my English isn’t too bad and i have been of some help.

(I’m a french citizen)

Yes, you can say :

On vient de recevoir un email d’Alyssa nous annonçant, tenez-vous bien, qu’elle aurait à son tour repéré un OVNI !

I think the begining of the sentence is a synonym of

Vous vous rendez compte, on vient de…

“Figurez-vous” and “tenez-vous bien” are two different expressions to express the need of attention from the interlocutor

But

Tenez-vous que

is not a correct. It means nothing.

PS : If you need more explaination, ask in comment, I will edit my answer.

When someone is about to announce some dramatic news, it’s not unheard of to request their counterpart to take a sit:

Assieds toi, s’il te plait. J’ai une nouvelle importante à t’apprendre.

The purpose is to avoid the counterpart to be hurt, if he or she come to faint when discovering this announcement.