One wouldn't.

One really, really wouldn't.

The two countries were at war; that precludes any kind of trade, immigration, or exchange of anything other than ammunition and the occasional package or correspondence proxied through the Red Cross (with that involving significant censorship at either end, generally limited to letters to and from POWs). It certainly means Canadian diplomatic and immigration offices, through which one had to apply to emigrate in the first place, were unavailable. In short, there was no one to apply to in the first place, and if they heard about this anyway the most likely assumption would be a particularly inept agent.

Generally speaking, immigration and emigration of almost any kind was completely shut down during the war. Canada admitted 8,504 immigrants in 1943, a lot of whom would have been from the US or from case-by-case situations like war brides, evacuee children who reached their majority and wished to stay in country, or other exceptional situations. For comparison, Canada admitted almost 105,000 immigrants in 1930 before the Great Depression, well, depressed immigration, and numbers gradually returned to that range in the years immediately following the war.

There were no boats to catch; any transatlantic-capable shipping was specific to the war effort, and passenger ships in particular were all pressed into service as troop- or hospital ships. As a result, anything afloat using the other side's flag was fair game; a hypothetical Romanian liner bound for Canada in 1943 would be attacked by any Allied ship, submarine or aircraft which noticed it between its home port and North America. (The Mediterranean, your character's most likely route of departure, would have been especially unpleasant this way.)

So that's the problem of getting there without being shot by the destination country or its numerous allies. What about being able to leave in the first place? Remember that Romania at this time was still fascist state aligned with the Axis - the coup through which the country switched sides didn't happen until the following summer - and it was very much engaged in a total war. Legal emigration wasn't a possibility, especially if the destination wasn't another Axis country for a good reason; in many countries at the time leaving one's job for another one was difficult enough. The Romanian government definitely wouldn't approve an exit visa at this time, and if someone requested one for an enemy country the most likely result would be a brief treason trial.

If, after all of that, someone managed to make it to Canada from Romania anyway, they would have been (at best) detained as an enemy alien for the duration of the war.

Not an ideal situation to try that.

So.

What are our friend's options in this case?

He had a couple, but they involved waiting out the war.

Late in the war, many civilians in central and eastern Europe decided it would be better to take their chances with the western Allies than the Soviet Union, and started fleeing in that direction once it became clear how things were going to end. Many Romanians, Hungarians, Poles, Germans (including Germans in other countries who didn't flee but were expelled), Baltic citizens, and others did this as part of an enormous flood of refugees trying with various degrees of success to stay clear of the Red Army. Several million wound up in displaced persons camps throughout central Europe when the fighting stopped (giving them the nickname of "DP" or "Displaced Person"). This often involved jumping through some hoops, as Allied authorities sought to make sure someone wasn't a war criminal and that they were really displaced by the war. For the most part the process of getting a spot in the camps was permissive, but it was also as unfeelingly bureaucratic as anything involving refugees is. People sometimes had to appeal a denial, pulling together references to prove they were who they said they were, or providing evidence that they wouldn't be safe back home (for example, if someone was in an anti-communist organization in Romania during the war). The United Nations Relief and Rehabilitation Administration, and from 1947 its successor, the International Refugee Organization, were very busy in the postwar years. The DP situation was a refugee crisis on - at least! - the same scale as the current Middle Eastern one.

Most spent several years there as situations gradually resolved themselves - this family might make their way back home, if home still existed; that family might settle with relatives in a nearby town; this other family might choose to put down new roots in occupied Germany, and so on. Many wanted to go home, or near home, but many others shared your character's mindset of "I'm done with this continent" and remains in the camps until an opportunity to leave arose. The camps weren't overflowing with milk and honey to say the least; things were crowded, often impersonal, and so on, but people generally had enough to eat and medical care. People with appropriate skills often found employment with the local occupation forces. For example, someone who was a mechanic before or during the war could spend time helping maintain cars and trucks local authorities used, sometimes getting certifications or character/employment references in the process. Overall, people who spent time in the DP camps don't speak fondly of them, but recognize that the circumstances were a lot better than what they experienced making it across war-torn Europe to one.

(As an aside: one set of displaced-persons records I worked on a couple of years ago involved a gentleman who spent two years working with a US Army maintenance battalion keeping their vehicles in order. The lieutenant supervising him gave him a letter to show employers before he left for Canada, certifying him as "capable of repairing and maintaining all U.S. Government vehicles." How's that for a job reference?)

They could apply for resettlement abroad in another Lots Of Bureaucracy process which could span a year or two while still in the camps. If they got through that, shipping was arranged to Canada; the DPs would sail usually from ports in Germany or the Netherlands, disembark in Halifax, Nova Scotia, and take a train from there to their arranged destination and employment. Their employment was arranged and not optional: part of the government's willingness to accept DPs was economic even before the assumption that they had debts to work off to the society. The large majority were required to spend at least one year in an arranged job, usually in natural resources (mining and forestry were very common, which was intensely frustrating for some. The mechanic I mentioned above had this happen to him). At the end of that time, they were free to do as they willed.

About 160,000 Europeans came to Canada through this general process between 1945 and 1950. They generally did well for themselves, although the "they're taking our jobs!" attitudes and general xenophobia were definitely a thing then. Many people used "DP" as an insult, and more traditional immigrants mentioned being confused for one to be an insult. It served as a solid dress rehearsal for the administration of other large refugee movements to Canada afterwards.

Your character's other most likely option was perhaps even harder: hunker down until the war ended, hopefully surviving the process, and either applying for emigration through the normal civilian channels (if Romania permitted this postwar) or fleeing to a neighboring country and then applying (if they didn't). These routes started opening up shortly after the war, especially as the displaced persons crisis wound down and conventional transatlantic traffic resumed in the late 1940s. This might have been difficult, since the Soviets sat on Romania pretty hard until the 1950s. A lot of Eastern Bloc citizens who came to Canada in the late-to-postwar years had to sneak across a border or two to do so.

Whichever route someone took to Canada, the path to citizenship was about the same. Assuming an immigrant kept his or her life in order, five years after the date of their arrival in Canada they were able to formally apply for Canadian citizenship.