Is it permissible to use a transportation card belonging to another person to benefit from a reduced student fare, if my card expired a short time before traveling, which does not justify purchasing a new card for a full month?

It is impermissible to use a card that you do not own, as this constitutes an infringement upon the rights of others and consuming their wealth unjustly. This is based on the Almighty's saying: "And do not consume your properties among yourselves unjustly," and the Prophet's (peace be upon him) saying: "Beware! Whoever wrongs a mu'ahid (a non-Muslim living under Muslim protection), or diminishes his rights, or burdens him beyond his capacity, or takes anything from him without his consent, then I will be his adversary on the Day of Resurrection." And his (peace be upon him) saying also: "Indeed, your blood, your wealth, and your honor are sacred to you, as sacred as this day of yours, in this month of yours, in this city of yours, until the day you meet Him... Indeed, the wealth of a person is not lawful [to take] except with his willing consent." The fact that you will use the card for one week does not justify the transgression. You must weigh the reduced price of the new card against the full price without a card.