Answer in brief
Do Catholics also have Assurance of Salvation?
No, Catholics have no certainty of their personal salvation, but they do have 1) an assurance that God intends to save humanity through his church, and 2) a hope that they may be among the saved.
At what point in their life can they say that they are saved and will go to Heaven?
When they have been faithful, they may be more assured of their salvation.
Or, do they strive their whole life in order to have Salvation?
Yes, Catholics believe one can never know for sure who is and is not saved, including themselves. They believe one must continue to be faithful and obedient in the hope that one day they will be saved.
The sixth session of the Council of Trent condemned those claiming an assurance of faith:
It is not to be said to any one boasting a confidence and certainty of the forgiveness of his sins, that his sins are forgiven, or have been forgiven; seeing this vain confidence, totally remote from piety, may exist in heretics and schismatics. ... As no pious man ought to doubt of the mercy of God, the merit of Christ, and the virtue and efficacy of the sacraments, so every one, while he beholds his own weakness and disinclination, may be in fear and dread respecting his own gracious state; seeing that no man can know with a certainty of faith, as to which there can be no lurking error, that he has obtained the grace of God. (Chapter 9)
Whosoever shall say that he holds it absolutely and infallibly certain that he shall have the great gift of perseverance even unto the end, if he has not learned this by special revelation, let him be anathema. (Canon 16)
Such people are guilty of the sin of presumption. On the other hand, one should hope in his own salvation, provided he has done good works:
Whosoever shall say that the righteous ought not, for the good works which may have been done in God, to expect and hope for eternal recompense from God through his mercy and the merits of Jesus Christ, if he persevere even to the end in well-doing and in keeping the Divine commandments, let him be anathema. (Canon 26)
Biblical basis against assurance
Ludwig Ott, in Fundamentals of Catholic Dogma (page 244), puts forward these verses as evidence in favor of the Council of Trent's declaration "against Calvin, that certainty in regard to one's predestination can be attained by special Revelation only":
Phil 2:12 Work out your salvation with fear and trembling. (cf. Ex 15:16; Jdt 2:28; Ps 2:11; Is 19:16)
I Cor 10:12 Whoever thinks he is standing secure should take care not to fall.
Other Catholics point out these verses:
I Cor 4:1-5 Thus should one regard us: as servants of Christ and stewards of the mysteries of God. Now it is of course required of stewards that they be found trustworthy. It does not concern me in the least that I be judged by you or any human tribunal; I do not even pass judgment on myself; I am not conscious of anything against me, but I do not thereby stand acquitted; the one who judges me is the Lord. Therefore, do not make any judgment before the appointed time, until the Lord comes.
The "special revelation" exception
Trent did not rule out that some might know by special revelation that they will be saved in the end.
The Catholic Church has ruled Lucia Santos' visions of Our Lady of Fatima as "worthy of belief." In one of them, Mary told Lucia that she would be taken to heaven "soon."
God only knows
According to Ott, in the same source previously quoted, our uncertainty of our predestination does not mean that God is uncertain. God knows, and has known, exactly who will be saved:
The Resolve of Predestination, as an act of the Divine knowledge and will, is as immutable as the Divine Essence Itself. The number of those who are registered in the "Book of Life" (Phil 4:3; Apoc 17:8; cf. Luke 10:20) is formally and materially fixed, that is, God knows and determines with infallible certainty in advance, how many and which men will be saved. What the number of the predestined is, God alone knows.
Perseverance to the end
Our uncertainty also doesn't mean that we can't be reasonably sure of our salvation; we just can't have an infallible assurance. After all, "The one who perseveres to the end will be saved." Matt 24:13 Ott lists a few things a person can do to be more sure of his salvation:
In spite of this uncertainty there are signs of Predetermination (signa praedestinationis) which indicate a high probability of one's predestination, e.g., a persevering practice of the virtues recommended in the Eight Beatitudes, frequent reception of Holy Communion, active love of one's neighbour, love for Christ and for the Church, veneration of the Mother of God.
Someone in a state of grace at their death can be said to have persevered to the end, while "the souls of those who die in a state of mortal sin descend into hell."