Eagle, lorica Segmentata was not developed to counter Falx, and they never used steel shields.
Anyway, Let's assumed that, Roman army was Imperial roman army (around the time of Augustus - Claudius), legion, or 4800 + cavalry + personnels= app. 5000. Samurai being Sengoku pre-gunpowder era.
Roman: 4800 legionaries, 120 cavalry, 60 Ballistae, 10 Onagri (Each century contains one ballista and its crew (they could be rota from century). Also a single Cohort has one onager. pl. Onagri)
Samurai: 2000 Ashigaru archers, 2000 Ashigaru pikemen, 1000 Spear Samurai. Also it contains numerous stone throwers (if to conpensate Roman artillery).
Before starting, there's a difference between the military philosophy. Roman army was designed to engage and destroy enemy formation, with very aggressive tactical tendency. Japanese, on the other hand, developed more passive way to fight. Their method is mostly shooting as many arrows and stones at first to disrupt enemy formation. It could be more accurate to say, they expected to break enemy with missiles. It is very rare to see Samurai fighting head-on. And this head-on style of battle is what Roman was made for.
Also formation. Roman legionary is formed to maximise frontal fighting ability. First lines have four Cohorts, behind them formes three Cohorts for follow-up/reinforcing, and then another three Cohorts for final reserve. Each Cohort was four rank deep, staggered, 120 men wide (appr.108m to 120m). Each Cohort's gap was open quite wide - to maximise manoevring. Cohort structure means that they can take any formation at will.
Samurai, on the other hand, formed for more all-direction defenciveness. They are divided into Sonae, which is of various size. Sonae was commanded by a powerful vessel under the lord (lord himself has his own Sonae). Unfortunately, not many example of Sonae was survived, but the best known one was virtually box. The outer ring was formed by missile troops who were made up of groups of five. Next came pikemen. Inner part was not box. two lines of spear Samurai. Behind them was the commander in the centre of guards in half-moon formation. This formation severely reduced the combat efficiency but it is virtually unflanked. And moreover, each line was a single line. Let's say they are made of 5 Sonae, 400 archers, 400 pikemen, 200 samurai which gives frontal length of 100m. All samurai are dismounted, except commanders of each squardron.
Fighting method has to be taken in account:
Legionaries are designed to fight in relatively close range, they use Scutum, Pilum (Heavy and light), Gradius, Pugio (dagger). Armour was, well, lets say Lorica Segmentata though it seems likely that many, if not majority, soliers wore Lorica Hamata too.
Samurai are more likely to fight in longer distance. Pikemen uses specially constructed pike, of 5 -6.3m. They are made of composite wood and bamboo reinforced with string and lacquer. It is flexible (actually bend slightly) and strong. It is said it can push back a charging horse without breaking. But, it's main usage is not thrust, but smash from above. They simply batter enemy to death. It can also be used to make makeshift lafts, ladders and stretch. Spear is usually 3.6m long, with head of various shape. Some had hooks in shaft. Katana is, in all aspects, sidearms and not really considered to be used in this simuration.
So, what might happen is... Total defeat of Japanese (draw was the best they can get)
1. Setting troops.
Japanese would place Sonae with either Crane Wing formation (Somae placed to make as wide front as possible) or Scale formation (like Roman would, or piling up one another). It really doesn't matter which one to choose. Outcome would be the same. Each Sonae's gaps are wide - perhaps 100m or more. they could hope that any troops went into these gaps are shot by flank archers.
Roman would make either traditional 4-3-3 formation (if Japanese took Scale formation), or more wider one, maybe 5-5 formation (for Crane Wing, each sonae would face one Cohort with another for back-up).
2. Initial contact 1.
Roman artillery fires upon Japanese Sonae. Since Sonae was very loose (just a collection of single lines), artillery fire would be not as effective as against other usual mates like gauls. Though Ballistae is pretty accurate and it could break the moral of enemy. Legionary advances.
3. Initial Contact 2.
As Legionary approached, they enter the arrow range, possibly about 50m. Because of its box-shape, Sonae's fire power is reduced to 1/4. of 400 archers in one Sonae, only 100 can shoot! Legionary's Scutum made it almost impossible to make effective hit.
At the same time, Hord of stone throwers running up and throw stones at them. It is how Jpanese battle starts. Not really effective against Roman. Japanese can hope that hail of stones and arrows disrupt enemy formation, or defeat them. Very unlikely.
4. Crucial Point.
Japanese has a decision here. Retreat (and make it draw), or fight (and suffer total defeat). Historically most of Japanese commander choose to retreat. This time, they choose to fight to show how Samurai dies.
5. Initial Engage.
Roman finally made to Japanese Sonae. Archers withdraw inside of Sonae, or somewhere and Pikemen raise their pike and ready for oncoming Roman.
Roman, as usual, throw pili into pikemen. 500 pili went to 100 pikemen (rest 300 are other sides of box).
Pikemen Ashigaru flee. 'Samurai's job is to die, Ashigaru's job is not to die. (Ashigaru Monogatari)' It is very likely that other Ashigaru and Samurai will flee as well, to end the game.
If pikemen held, their battering is easily dealt with Sctum and Legionare will make short work of pikemen.
6. Legionare vs Samurai
If Legionary kept one pilum, it is likely to cause the volley to flee the first line of Samurai. If not, they would enter hand-to-hand. Legionary would find difficult to get into the range. And Japanese line is easily move backwards (it's just a single line, after all). However, since this very thin line means they tend to shrink rapidly when they starting to get casualty. Also thin line deteriorate morals in combat.
Roman attempt to make flank manoeuvre is counterd by flank archers and pikemen and as in previous sections, they will be destroyed and encircle enemy Samurai.
If they get close, winning fomura of Scutum+Gladius equals victory. Katana has great difficulty getting around of Scutum and shield bashing. Legianry has a lot of experience fighting these type of enemy.
7. End
It is very unlikely that Samurai would stay till this phase, more likely they will flee before contact. If they stood firmly, Samurai would be killed to the last man and add yet another glory to the Roman Empire.
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