Gosh, what a big question! One could write several books on Roman and Byzantine religious practices at either time point. I can speak to some of the liturgical differences, but not much else. For simplicity, I'm going to assume that your Roman citizen is either from Rome itself or Constantinople.

The central rite of all apostolic churches is the Eucharist, based in part on a commemoration of the Last Supper between Christ and his apostles. The service in which this is celebrated is known either as the Mass (in the Western, primarily Catholic Church) or Divine Liturgy (in the Eastern, primarily Orthodox Church). Both are derived from the earliest known Christian rituals; our best contemporary descriptions come from the Didache and writings in Greek by Justin Martyr and Clement of Rome from around 100 CE. You can see from Justin Martyr's Apology that the central prayers of the Eucharist are already in place, along with intercessionary prayers and the exchange/kiss of peace. Additionally, Clement in Chapter 34 of his Epistle describes a prayer very similar to the Sanctus, found in both the Mass and Divine Liturgy. These building blocks are present in the liturgies of East and West in 400 CE, 1400 CE and today.

Early church centres adapted and developed this basic liturgy in different ways. The two main branches are that of Rome (the Roman rite) and that of the Eastern churches (Jerusalem and Antioch, followed by Constantinople). By 400 CE, many of the liturgical differences between the Roman rite and Constantinople's rite observable in 1400 CE (and indeed today) were already in place.

We do not know the exact form of the rite used in Rome in 400 CE. Our earliest codification comes from Pope Gregory in the sixth century CE. However, we can piece some of the Mass of 400 CE together from letters between bishops of the time and surviving priests' sacramentaries (basically crib sheets). For one, we know from Augustine of Hippo, a contemporary bishop, that it was called the Mass! Most of the Roman liturgy (minus some prayers like the Kyrie eleison, which remain in Greek today) was celebrated in Latin rather than Greek; this change had begun in the third century and was more or less complete by 400 CE. Additionally, by 400 CE Jerome had finished his systematic translation of the Old and New Testaments into Latin (the Vulgate). This translation was still in use for readings in 1400 CE, and is still used with minor revisions to this day in the normative Latin versions of the Ordinary and Extraordinary forms of the Catholic Mass. The centerpiece of the liturgy, the Eucharistic prayer, was almost certainly some form of the Roman Canon in 400 CE. This continues as the normative Eucharistic prayer in the Ordinary and Extraordinary forms of the Roman Rite today. Certainly, all the Roman martyrs referred to in the Canon were dead by 400 CE. However, some parts were definitely missing from the mass of 1400 CE in 400 CE: one big example is the Agnus Dei prayer, sung after the words of institution (where, according to the Roman church, the bread and wine become the body and blood of Christ), which was not added until the seventh century CE. The Roman Canon in 400 CE may or may not have had an epiclesis, where the priest invokes the Holy Spirit over the Eucharistic bread and wine. This is present in early rites, and is seen as a crucial part of Byzantine services (the Orthodox believe this is where the bread and wine become Christ's body), but is missing from the Roman Canon by the seventh century. Your Roman may also be caught out by the kiss/exchange of peace; this was moved in the service at some point between the third and seventh centuries from before the words of institution in the Eucharist to after the recitation of the Pater noster, or Lord's Prayer, nearer the end of the service.

Likewise, we do not have a codification of the Byzantine rite of 400 CE; our best guess is that it was similar to the modern Divine Liturgy of St Basil, who lived in the fourth century CE. Versions of this ancient rite are used in the churches of Alexandria, Antioch and Constantinople today. The Bishop of Constantinople in 400 CE was John Chrysostom, who modified this liturgy significantly; his version is now the normative liturgy of the Orthodox Church. Either service would certainly have been celebrated in Greek in the Constantinople of 400 and 1400 CE, with readings in Greek from the Septuagint Old Testament and a New Testament based on the Byzantine Codex. Both of these are still in use today in Greek-speaking churches.

All of this is a very long winded way of saying that liturgy-wise, most of the differences between Rome and Constantinople were already in place by 400 CE. Hence your citizen would feel more out of place in one or the other depending on what end of the Empire he was from. If he was from the Eastern Empire, he would be able to follow the general form of the liturgy in Constantinople both in 1400 CE and the present day. If he was from Rome, he would probably be able to recognise what was going on in a Roman service of 1400 CE, though some parts would be unfamiliar or in the wrong order. However, like worshippers today, he'd feel very out of place if swapped from Rome to Constantinople, or vice versa, without warning.

Let’s assume you mean Christianity. Most likely Western Europe, which still ascribed itself to the “Roman” brand of Christianity.

After the “Fall” of Rome, (I use quotation marks, because the historical consensus today is that Rome did not truly), there were three empires that formed out of it. The Islamic World, Byzantium, and Western Europe - we’ll call them the sibling heirs. Each of them retained certain characteristics of Rome - hell, Byzantium stiles called itself the Roman Empire - but there were distinct differences in religion that served to really separate the sibling heirs as the Middle Ages went on. The Islamic World is not involved in your question, so let’s take a look at Byzantium and Western Europe.

Byzantium in particular deviated from Roman Christianity early with the rise of icons and their worship c.680 CE, though eventually Emperor Leo outlawed icons in the 700s CE. But, icons rose to popularity again in the 800s. At this time it should be noted that Western Europe was not wholly Christian, and Constantinople held control of Rome, and the Emperor held most of the power in the church. Tensions rose between the pope and Byzantium multiple times through the late 700s and early 800s, until Pope Stephen II appealed to Pippin III - king of the Merovingians - for aid. That is when Rome and the Roman version of Christianity fell fully under the sphere of the Western Europeans. (You can see here the beginnings of the divide between the modern day Orthodox and Catholic churches).

Charlemagne’s conquests is then largely responsible for the conversion of many parts of Western Europe to Roman Christianity.

While the Christianity that was practiced in Byzantium or Western Europe was exactly like its predecessor, the most similarities would be seen in Western Europe.

Edit: If you would like to find more information on this, Barbara Rosenwein’s textbook A Short History of the Middle Ages provides a fairly comprehensive (if dry) look at the divergence between Byzantium and Western Europe in regards to religion. Some primary sources like the Pactus Legis Salicae, The Life of Charlemagne, and Charlemagne’s Admonitio Generalis show the power shift of Rome from Constantinople to Western Europe and how this affected daily life.