Magic weapon will work against the lich

Regarding the last sentence specifically, magic weapon does not affect an area, it only affects a target (an object). Thus, this sentence has no relevance to this spell. The only purpose of this sentence is to exclude the globe from area of effect spells like fireball and the like.

Magic weapon affects one target: a nonmagical weapon that you touch. That weapon then becomes a +1 magic weapon for the duration. When you then try to use the weapon against the lich you are not affecting the lich with the spell, you are affecting them with the newly enhanced weapon. Thus, nothing about doing this would be blocked by globe of invulnerability. The same is true if you cast enlarge or invisibility or polymorph on yourself and moved into the globe to attack.

If globe of invulnerability did work against magic weapon it would lead to ridiculously powerful results such as creatures under the invisibility suddenly become visible since the spell "affects" the lich by way of making the PC impossible for them to see. Or that mirror image stops working because it "affects" the lich by making it hard for them to hit the PC. In essence it would make this spell on par with antimagic field, an 8th level spell.

Slightly Ambiguous, But Previously Cast Spells Probably Still Work

The crux of the matter here is that the word "cast" typically is the same in the past and present tense. As such, the precise wording of globe of invulnerability stopping spells "cast from outside the barrier" could intend to refer to past, present, both, or exclusively one (for example, only spells cast before the globe existed), and it would have exactly the same grammatical structure.

We have guidance, however, in the other sentences of the spell's description. We are told (PHB, p. 245, bold added):

Such a spell can target creatures and objects within the barrier, but the spell has no effect on them.

This sentence uses terms exclusively in the present tense. It doesn't say that the spell can "have targeted" creatures (indicating that they targeted them in the past and are still targeting them), but rather speaks of only spells that are now attempting to target a creature once the globe of invulnerability is in effect. As such, it is likely that both the intended timing of the targeting and being affected are the same: meaning the globe of invulnerability blocks spells against something inside the globe when the other spells are cast.

The other interpretation leads to madness

The grammatical ambiguity permits people to interpret this spell as they wish. But keep in mind that many complex cases could arise if globe of invulnerability blocked previously cast spells. For example, let's say a wizard casts mirror image on themselves, then moves 20 feet and (in the next round) casts globe of invulnerability. Was the spell mirror image cast from outside the globe or not? Such a question is needlessly open to interpretation (can you be said to be "outside" the globe when it doesn't exist? Since the globe appears around you, is the source of mirror image you, or your previous location?), but would not even come up if the globe only blocks subsequent spells.