RadagastyI'm not from Cambridge, so I'm not offering any course-specific advice, but I come from a background in historical and comparative philology, so I may be able to offer some advice on subject choice.
You've settled on Old English, which is an obvious choice, and, to match it, I would recommend Insular Latin. Here, I am assuming that you don't already have a background in Classical Latin. If you do, it would be fairly easy to pick up on Insular Latin, so choose another subject instead. I cannot emphasise too much the importance of Latin in this field. It is absolutely essential if you wish to pursue further studies or research in the field, for there are many mediaeval documents written in Latin that are relevant to ASNC. If at all possible, I would strongly advise you to learn Modern German, at least to a rudimentary level, perhaps through the Cambridge Language Centre.
Note that the above advice is given without knowing which particular field of ASNC you wish to specialise in. If it is Anglo-Saxon studies, give more weight to Scandinavian topics (i.e., Scandinavian history, Old Norse) over Celtic topics. Clearly, if you wish to specialise in Celtic studies, then Mediaeval Welsh and Irish would most be useful, as well as Britonnic and Gaelic history. Here, the acquisition of Latin is less important, but still recommended.
Palaeography is less important than Latin, but no less essential if you plan to undertake research in the area. Any non-trivial study of the field would most likely require you to be able to read mediaeval manuscripts, and studying palaeography would help immensely. Note tha palaeography isn't simply the study of old handwriting: it is a broad field covering many topics like manuscript conventions, textual authenticity, etc.
I wouldn't worry too much about the languages going pear-shaped too much. Latin is taxing on the memory, but, in many ways, Insular and Mediaeval Latin are more straightforward than classical Latin, and shouldn't pose too much of a problem. Old English is quite easy to learn, although the syntax can be quirky.
The main problem you're going to encounter is variation: variation through time and space, i.e., regional variation in Old English and its evolution through time, as well as internal variation, i.e., arbitrary and non-significant differences in spelling and grammar, etc., that may be present in a single document, often due to lack of standardisation. These variations can be most frustrating when trying to decipher a document, especially if it is a manuscript text in illegible handwriting. At the beginning, though, you would most likely be reading redacted texts where spelling has been regularised, etc.
My recommendation, then, or rather, the subjects I would choose in your position, are:
Old English
Insular Latin
palaeography
Anglo-Saxon history
Scandinavian history
Old Norse
If you feel you won't be able to cope with a third language, then perhaps substitute another history (I would go for Brittonic over Gaelic here, though the latter is perhaps more relevant to-day) for Old Norse.
Keep in mind that I am not from an ASNC background, but rather one of philology (historical and comparative) and history (late antiquities and early middle ages), so my advice may be a little biased.