Dua and supplication are the needful connection between the servant and his Lord for the fulfillment of worldly and otherworldly needs. Dua for oneself and others is always something good and entails great blessings.
Praying for oneself on Ashura is okay, as a matter of fact, there are some special prayers and supplications for this day, duas and ziyarat being some of them, and in most if not all of them, one prays for oneself as well.
Let it not remain unmentioned though that it is appropriate for one to pay less attention to worldly matters on this day and only as much as needed, and to make this day a day of mourning and tears and tragedy.
Dua and supplication are the needful connection of the servant and his Lord for the fulfillment of worldly and otherworldly needs. It isn't true that one shouldn’t pray for oneself on the day of Ashura, because there are specific things the ahadith have recommended that we do on this day, duas and ziyarat being some of them; in these duas and ziyarats one prays for oneself as well.
For example, in the famous ziyarat of Ashura, there are some segments in which the reciter prays and asks Allah for himself:
“اللَّهُمَّ اجْعَلْنِی عِنْدَکَ وَجِیها بِالْحُسَیْنِ علیه السلام فِی الدُّنْیَا وَ الْآخِرَة”
(O Allah! Through Husein, make me stationed and distinguished in your eyes in this world and the next).
“اللَّهُمَّ اجْعَلْنِی فِی مَقَامِی هَذَا مِمَّنْ تَنَالُهُ مِنْکَ صَلَوَاتٌ وَ رَحْمَةٌ وَ مَغْفِرَةٌ اللَّهُمَّ اجْعَلْ مَحْیَایَ مَحْیَا مُحَمَّدٍ وَ آلِ مُحَمَّدٍ وَ مَمَاتِی مَمَاتَ مُحَمَّدٍ وَ آلِ مُحَمَّد”
(O Allah! Make me in the place and station where I stand, of those whom you bestow with your blessings, mercy and forgiveness; O Allah! Make my life [like] the life of Muhammad and the progeny of Muhammad and my death, [like] the death of Muhammad and the progeny of Muhammad).
Also, there is a prayer and ensuing dua that has been introduced as one of the things to be done on this day by Imam Sadiq (as). In many of its segments, one asks Allah for forgiveness and mercy and the fulfillment of his wishes and needs.
Therefore, praying for oneself on Ashura is totally okay, but generally, the infallibles have strongly recommended that we pray for others as well and have said that it shouldn’t be that the believer only prays for himself and forgets his brothers.
As has been mentioned in a hadith by Imam Sadiq (as): “The messenger of Allah (swt) said: Whosoever says: اللهم اغفر للمؤمنین و المؤمنات (O Allah forgive the believers of men and women), Allah (swt), will write him a good deed, erase a sin and raise him a degree (spiritual degree) for every believer He has created and will create from the time of Prophet Adam (pbuh) till the Day of Judgment.
It is noteworthy that there are hadiths that have been transmitted to us that inform us of the need to spend this day in mourning and grief, and to refrain from worldly matters as much as possible and to only engage in them minimally, and that one’s activities in this day should differ from other days, because the Umayyids looked at this day as a blessed one and saw it as their day of victory. But this doesn’t mean that one should refrain from the necessities of life on that day.
Hadith: Imam Ridha (as) says: “Whosoever tries to put aside his needs on the day of Ashura, Allah (swt) will fulfill all of his worldly and otherworldly needs, and for whomever this day is a day of grief and tragedy and weeping, Allah (swt) will make the Day of Judgment a day of happiness and joy, and whoever sees this day as a blessed one and stocks up on it, Allah (swt) will not bless what he has stored and he will be resurrected on the Day of Judgment with Yazid and Ubeydullah bin Ziyad and Umar bin Sa’d in the lowest levels of Hell.”
In any event, as was said, refraining from worldly matters on this day has nothing to do with not praying and making dua for oneself and refraining from the necessities of life.
For further information, see:
1. The conditions for the certain answering of duas, Question 197 (website: 983).
2. The relationship between dua and effort, Question 6154 (website: 6286).
3. The philosophy behind dua, Question 8961 (website: 9037).
 Majlisi, Muhammad Baqir, Bihar al-Anwar, vol. 98, pp. 303-307, Al-Wafa Institute, Beirut, 1404 (AH); Sheikh Tusi, Misbah al-Mutahajjid, pp. 782-787, Fiqh al-Shia Institute, Beirut, 1411 (AH).
 Bihar al-Anwar, vol.90, pg. 391: “مَن قَالَ اللَّهُمَّ اغْفِرْ لِلْمُؤْمِنِینَ وَ الْمُؤْمِنَاتِ کَتَبَ اللَّهُ لَهُ بِکُلِّ مُؤْمِنٍ خَلَقَه اللَّهُ مُنْذُ خَلَقَ اللَّهُ آدَمَ إِلَى أَنْ تَقُومَ السَّاعَةُ حَسَنَةً وَ مَحَا عَنْهُ سَیِّئَةً وَ رَفَعَ لَهُ دَرَجَةً”
 Sheikh Hurr Ameli, Wasa’el al-Shia, vol. 14, pg. 504, Aal al-Bayt Institute, Qom, 1409 (AH).