jemandem/etwas gegenüberstehen
to face
To be confronted with a situation, problem, or challenge.
Wir stehen einer großen finanziellen Herausforderung gegenüber.
We are facing a major financial challenge.

to face / to stand opposite
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to face
To be confronted with a situation, problem, or challenge.
Wir stehen einer großen finanziellen Herausforderung gegenüber.
We are facing a major financial challenge.

'Gegenüberstehen' (to face / to stand opposite) is a separable verb. In a main clause, the prefix 'gegenüber' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Er steht dem Projekt kritisch gegenüber.
He is critical of the project.
This verb always requires an object in the dative case. You stand opposite or face 'jemandem' (someone) or 'etwas' (something), not 'jemanden' or 'etwas' in the accusative.
Der Angeklagte stand dem Richter gegenüber.
The defendant stood opposite the judge.
In the physical meaning, the verb is often used reflexively: 'sich gegenüberstehen' (to stand opposite each other). This emphasizes that two or more people or things are looking directly at each other or are positioned in front of each other.
Die beiden Mannschaften standen sich vor dem Spiel auf dem Feld gegenüber.
The two teams stood opposite each other on the field before the game.
When the verb describes a confrontation, the dative object is usually an abstract challenge. Typical words are 'Problem' (problem), 'Aufgabe' (task), 'Herausforderung' (challenge), or 'Entscheidung' (decision).
Das Unternehmen steht großen wirtschaftlichen Problemen gegenüber.
The company is facing major economic problems.
To express an attitude, the verb is almost always combined with an adverb that describes the opinion. Typical adverbs are 'skeptisch' (skeptical), 'kritisch' (critical), 'positiv' (positive), or 'offen' (open).
Neuen Technologien stehe ich grundsätzlich offen gegenüber.
I am generally open to new technologies.
to encounter / meet
'begegnen' and 'gegenüberstehen' can both describe dealing with a problem, a difficulty, or an attitude. 'begegnen' tends to emphasize how you react to it or how you treat someone. 'gegenüberstehen' tends to emphasize the situation: you are confronted with something or have a particular opinion about it. Learners may confuse the words because both are often used with the dative and appear in similar topics. Put practically: you stand facing a problem ('einem Problem gegenüberstehen'), but you face or address the problem with calm or courage ('dem Problem mit Ruhe oder Mut begegnen').

Wegen mehrerer Termine stehen wir heute einem engen Zeitplan gegenüber.
Because of several appointments, we are facing a tight schedule today.
Frau Schwarz begegnet den neuen Aufgaben mit großer Ruhe.
Ms. Schwarz approaches the new tasks with great calm.