Did Jesus Christ promise to St. Faustina that if people make a good confession and receive Holy Communion at mass on the second Sunday of Easter they will have a fresh start all over again?
The short answer is yes.
There are two things to be considered in this question. What are the promises made by Our Lord to St. Faustina in regards to Divine Mercy Sunday and what are the indulgences that the Church has granted for the Sunday following Easter.
To ensure that the faithful would observe this day with intense devotion, the Supreme Pontiff himself established that this Sunday be enriched by a plenary indulgence, as will be explained below, so that the faithful might receive in great abundance the gift of the consolation of the Holy Spirit. In this way, they can foster a growing love for God and for their neighbour, and after they have obtained God's pardon, they in turn might be persuaded to show a prompt pardon to their brothers and sisters.
Three conditions for the plenary indulgence
And so the Supreme Pontiff, motivated by an ardent desire to foster in Christians this devotion to Divine Mercy as much as possible in the hope of offering great spiritual fruit to the faithful, in the Audience granted on 13 June 2002, to those Responsible for the Apostolic Penitentiary, granted the following Indulgences:
A plenary indulgence, granted under the usual conditions (sacramental confession, Eucharistic communion and prayer for the intentions of Supreme Pontiff) to the faithful who, on the Second Sunday of Easter or Divine Mercy Sunday, in any church or chapel, in a spirit that is completely detached from the affection for a sin, even a venial sin, take part in the prayers and devotions held in honour of Divine Mercy, or who, in the presence of the Blessed Sacrament exposed or reserved in the tabernacle, recite the Our Father and the Creed, adding a devout prayer to the merciful Lord Jesus (e.g. Merciful Jesus, I trust in you!");
A partial indulgence, granted to the faithful who, at least with a contrite heart, pray to the merciful Lord Jesus a legitimately approved invocation. - Indulgences attached to devotions in honour of Divine Mercy
Here are the other main differences between the "special graces" promised by Jesus for Divine Mercy Sunday, and the plenary indulgence offered by the Church for special devotions to The Divine Mercy on Mercy Sunday:
The special graces that our Lord promised for Mercy Sunday come solely through the reception of Holy Communion on that day, in a state of grace, with trust in The Divine Mercy. Any plenary indulgence granted by the Church, on the other hand, involves the fulfilment of a number of conditions, including prayer for the pope's intentions, confession and Holy Eucharist, and the carrying out of the special indulgenced work (in this case: participating in public devotions to The Divine Mercy on Mercy Sunday itself).
The special graces promised by our Lord for Mercy Sunday can be received by a soul in a state of grace, but with imperfect love for God, and imperfect contrition for sin—as long as the soul merely trusts in the Mercy of God, and clings to Him because of His promised benefits. A plenary indulgence, however, can only be obtained through the performance of an indulgenced work as an expression of pure love for God. If the intentions of one's indulgenced work are not pure (say, the work is done in part out of fear of hell or purgatory, and loathing for oneself — or the desire to impress one's friends and relatives! — rather than purely out of love for God) then the indulgence obtained will be only partial, not plenary.
The special graces that our Lord promised for Mercy Sunday can only be received for oneself, through the devout reception of Holy Communion on Mercy Sunday (as stated in #1 above). A plenary indulgence, however, can be offered for oneself, or for souls suffering in purgatory.
The most special grace promised by our Lord for Mercy Sunday is nothing less than the equivalent of a complete renewal of baptismal grace in the soul: "complete forgiveness (remission) of sins and punishment." Such a measure of grace can only be received otherwise through baptism itself, or through making a sacramental confession as an act of perfect contrition for sin, out of pure love for God.
As you can see, this is no easy topic. However, you can find a more in-depth treatment of this same subject in the John Paul II Institute of Divine Mercy section of this same website, in the document entitled "Understanding Divine Mercy Sunday." - Mercy Sunday's Special Graces, Plenary Indulgence: Are They the Same?
According to St. Faustina, Our Lord promises to those who go to confession and communion on this day, the remission of the guilt and the punishment of sins.
On that day the very depths of My tender mercy are open. I pour out a whole ocean of graces upon those souls who approach the fount of My mercy. The soul that will go to Confession and receive Holy Communion shall obtain complete forgiveness of sins and punishment. On that day all the divine floodgates through which grace flow are opened. Let no soul fear to draw near to Me, even though its sins be as scarlet. My mercy is so great that no mind, be it of man or of angel, will be able to fathom it throughout all eternity. (Diary of Sr. Faustina, 699) - Mercy Sunday or Feast of Mercy