In the construction "X is relieved from Y", relieved acts as a verb and signifies that the “burden” of Y is lifted off of X. The same thing is expressed with the preposition "of".
The commanding officer was relieved from duty.
An unexpected inheritance relieved William of his debts.
In the construction "X is relieved at Y", relieved acts as an adjective and signifies that X feels a ‘sense of relief’, a sense that a burden of anxiety has been lifted off of him, because Y happened. That is the case in your headlines:
Seniors feel a sense of relief because the community center gained a 'reprieve' (presumably the center was going to be closed, but the closure was temporarily cancelled).
Egypt feels less anxious because it has been decided that crowds will not be banned in the Ghana World Cup qualifier.