Your question was addressed in the following paper:
Carmen Elvira-Donazar and Luis-Javier Hernandez-Patricio. Closed model categories for the $n$-type of spaces and simplicial sets. Math. Proc. Camb. Phil. Soc. (1995), 118, 93.
Allow me to define an $n$-fibration by quoting from the introduction: Let $I^p$ be the $p$-dimensional unit cube, $V^{p-1}$ be the union of all faces of $I^p$ except for $I^p\times \{1\}$ and $\partial I^p$ the boundary of $I^p$. A map $f$ is an $n$-fibration if it has the right lifting property with respect to $V^{p-1}\to I^p$ (for $0 < p \leq n+1$) and with respect to $V^{n+1}\to \partial I^{n+2}$.
With this definition, and your notion of an $n$-equivalence they prove that $Top$ (meaning a suitable cartesian-closed version) is a model category. So you can forget all mention of cofibrations and get the fibrant-object structure you wanted. The proof proceeds by way of Simplicial Sets, so if you read that paper you'll probably learn loads more about $n$-fibrations. For instance, Corollary 2.1 says trivial $n$-fibrations are exactly maps which have the RLP with respect to $\partial I^p\to I^p$ for $0\leq p\leq n+1$.
It is not difficult to see from the description of $n$-equivalences and $n$-fibrations that in the limit as $n\to \infty$ you get the usual model structure on $Top$. I should mention that this paper of Golasinski and Goncalves credits this model structure to Tim Porter and J.L. Hernandez via Categorical models of $n$-types for procrossed complexes and $\mathcal{J}_n$-prospaces from the 1990 Barcelona Conference on Algebraic Topology. But I couldn't find an online copy of that, so I went with the reference above instead.
Note that the dual question to your question (declaring $X\to Y$ to be an $n$-equivalence if $\pi_k(X)\to \pi_k(Y)$ is an isomorphism for all $k>n$) has also been answered, and again there is a model structure. Here is a reference:
J. Ignacio Extremiana Aldana, L. Javier Hernández Paricio, and M. Teresa Rivas Rodríguez. A closed model category for ($n-1$)-connected spaces. Proc. Amer. Math. Soc. 124 (1996), 3545-3553