In “Manfred” what ideas or concepts show continuities between the Romantic Era (Romanticism) and the modern age?

The main way in which Manfred anticipates modernism is in its focus on the interior life its main character. While Byron does not “psychologize” Manfred in any real sense, Manfred’s internal struggle is similar to the alienation we find in the moderns.
Romanticism and modernism both share a concern for the individual, but differ in the ways they construct the individual and in the social and political forces they were a reaction against. To give a very crude sketch, Romanticism was a reaction against the Enlightenment; it asserted the primacy of emotion over reason, and valued democracy and individual rights over the rights of monarchs. The Byronic hero, of which Manfred is the great example, is typically a mysterious figure, alone, suffering from a some terrible event in his past, who stands apart from conventional society and lives by his own moral code. Modernism, on the other hand, conceives of the individual in psychological terms; as a response to the rise of capitalism, the individual is often alienated from other people and institutions. If the Byronic hero struggles against Nature to assert his own personality, then the “Modernist hero“ struggles internally against his own self doubts and the broken social and personal relationships that mirror the alienated nature of modern society as a whole. Think of T.S. Elliot’s Prufrock as a kind of modernist counterpart to Manfred in this regard.
If we think about Manfred in a modernist context, we are forced to reinterpret some of Byron’s major plot points, and perhaps change our view of Manfred as a character. Manfred’s mysterious sin against Astarte, for example, can be understood less as a crime abhorred by nature and more in terms of a personal failure on Manfred’s part; the forgetfulness he seeks, in this view, is not so much a call for divine forbearance but an expression of personal guilt. When Manfred contemplates suicide in Act II, he soliloquizes

There is a power upon me which withholds,And makes it my fatality to live,—If it be life to wear within myselfThis barrenness of Spirit, and to beMy own Soul's sepulchre, for I have ceasedTo justify my deeds unto myself—

The supernatural “power” the dooms Manfred to become his “own Soul’s sepulchre” becomes, in modernist terms, his own internal misgivings about his actions. Although expressed in a more grandiose way, Manfred’s inaction (and his reasons for inaction) were expressed more concisely (if prosaically) by Elliot when he has Prufrock exclaim,

I grow old... I grow old...I shall wear the bottoms of my trousers rolled.

Prufrock’s fear of aging, and his heightened concern for the ridiculous, mirror Manfred’s inability to draw inspiration from the beauty of the mountains; both characters can be understood as alienated from the thing they love. This is made explicit in Manfred by the advent of the Chamois Hunter, who saves Manfred and serves as a point of comparison. The Chamois Hunter, living apart from human society in the mountains, represents a kind of “pure” existence in harmony with Nature; in modernist terms, because the Chamois hunter is somewhat isolated from capitalism, he does not suffer from the alienation Manfred (and Prufrock) do.