Can Mass be celebrated at any hour of the day?

Based on the summarized missal info on the USCCB's website, it seems like the Easter Vigil, at least, can take place any time between sunset and sunrise.

With respect to your quote from Eucharistic Prayer III: I think "from the rising of the sun to its setting" is meant in a geographical sense, not a time-of-day sense. The previous official translation of the prayer rendered it as "from east to west".

See the § "De Hora celebrandi Missam" ("Regarding the time of celebrating Mass") of the 1957 Missale Romanum:

Missa privata saltem post Matutinum et Laudes ab una hora ante auroram usque ad unam horam post meridiem dici potest.
_{A private Mass can be said at least after Matins and Lauds from one hour before sunrise until 1:00 PM.}

2. Missa autem Conventualis et solemnis sequenti ordine dici debet. In Festis Duplicibus et Semiduplicibus, in Dominicis, et infra Octavas, dicta in Choro Hora Tertia. In Festis Simplicibus et in Feriis per annum, dicta Sexta. In Adventu, Quadragesima et Quatuor Temporibus, etiam infra Octavam Pentecostes, et Vigiliis quae jejunantur, quamvis sint dies solemnes, Missa de Tempore debet cantari post Nonam.
   _{But a Conventual solemn Mass should be said according to [the rules of] the [religious] order. On Double and Semi-double feasts, Sundays, and within Octaves, it is said in Choir during Terce. In Advent, Quadragesima, and Ember Days, even within the Octave of Pentecost, and Vigils in which one fasts, even if they are solemn days, the Proper of the Season should be sung after None.}

3. Missa autem Defunctorum dici debet post Primam diei: ubi vero dicuntur eorum Vigiliae mane post Matutinum diei, dictis hujusmodi Vigiliis cum Laudibus, immediate dici potest Missa pro Defunctis. Sed in die Commemorationis Omnium Fidelium Defunctorum Missa Conventualis dicitur post Nonam, quia eo die est principalis. In die autem Depositionis Defuncti, vel Tertio, Septimo, Trigesimo aut Anniversario solemni, in quo fit concursus populi, poterit similiter dici ultimo loco post Nonam.
   _{But a Mass of the Dead [requiem] should be said after Prime of the day; but where Vigils are said early after Matins of the day, so-called "Vigils of Praise," a requiem can immediately be said. But on All Souls' Day [Nov. 2], the Coventual Mass is said after None, because that day takes precedence. On a day of Burying the Dead, or on the Third, Seventh, or Thirtieth day or solemn Anniversary, in which the congregation is present, it will similarly be said last after None.}

4. Excipiuntur ab hoc ordine dicendi Missam Conventualem, Missae in Nativitate Domini, quarum prima dicitur post mediam noctem, finito Te Deum laudámus in Matutino: secunda in aurora, dictis Laudibus et Prima: tertia vero in die post Tertiam, vel ubi aliter ex dispensatione Apostolica disponatur.
   _{Exceptions to this order of saying Conventual Masses are the Masses of the Nativity of the Lord [Christmases], the first of which is said after midnight, finishing with the Te Deum laudámus at Matins; the second at sunrise, having said Lauds and Prime; the third during the day after Terce, or at another time with an Apostolic dispensation [bishops' dispensation].}

5. Missae Votivae, quia non correspondent Officio, si solemniter celebrentur pro re gravi, vel publica Ecclesiae causa, cum populi frequentia, dicantur post Nonam.
   _{Votive Masses, because they do not have a corresponding Office, if solemnly celebrated for a serious reason, or for a public cause of the Church, with the congregation present, are said after None.}

Can Mass be celebrated at any hour of the day?

The short answer is: Yes, if there is a valid reason for it.

Masses can be said at any time day or night if there is a real necessity or spiritual need for it.

This answer will be based more on actual experience than with concrete sources. For some reason rubrics on this point are quite hard to locate for those who follow the Ordinary Form of the Mass (OF). Geremia gives a good answer, but the references treated are in reference to the Extraordinary Form of the Mass (EF). I can not locate any guidelines for the New Mass.

Yet the rubrics that Geremia lays out could also be altered by some real need, circumstance or local custom.

Under certain circumstances rubrics may be modified by custom, but in this respect they do not differ from laws in general. - Rubrics

I spent some 20 years affiliated with religious communities that say both forms of the Mass used in the Latin Rite. It is not uncommon for a priest in a religious community to say Mass extremely early because of the necessity of driving a great distance that day. This applies to both Forms of Mass of the Roman Rite.

New Years Day may be celebrated at midnight too. We had Mass celebrated by an Abbot in the Extraordinary Form on New Years Day 2000 as a way of welcoming in the New Millennium. I am sure some priests still do this.

In the monastery I stayed in for several years, one priest was assigned the task of saying Mass at 4:00 AM, so that the Brothers could be free to go and milk the cows. They called this the Farm Mass.

I can not tell you how many times I served a Mass (of either Form) between 2:00 AM and 5:00 AM because a priest (Religious or secular) had to leave on a long trip on that particular day.

Thus Mass may be celebrated at anytime day or night, if there is a valid reason for doing so. And that goes for both Forms of the Mass (OF and EF) in the Latin Rite.